MGF1106

The symbol for n factorial is​ _______.

11.7 n!

Evaluate the expression 9P0.

11.7 1

3.2 The only way a disjunction ( p v q ) is false is when

both p & q are false

Are S' ∩ T and (S ∪ T')' equal​ statements?

YES

8.1 Two​ angles, the sum of whose measures is 90°​, are called

complementary angles For​ example, angles ∢A and ∢B given below are complementary because 25°+65°=90°.

8.2 If the corresponding sides of two similar figures are the same​ length, the figures are called​ _______

congruent figures.

If the probability that an event occurs is 0.6​, the probability that the event does not occur is​ _________.

11.1 .4

Select the correct choice that completes the sentence below. For any event​ A, ​P(A)+​P(not ​A)=

11.1 1

Probability determined by the relative frequency of occurrence of an​ event, or actual observations of an experiment is called​ _______ probability.

11.1 Empirical

A six​-sided die is tossed. Determine the odds against rolling a 5.

11.2 5:1

There is a rack of 11 billiards​ balls, numbered from 1 to 11. If one ball is selected at​ random, determine the odds in favor of it being an odd​-numbered ball.

11.2 6:5

If the odds of a horse winning a horse race are 2:7​, then the odds against that horse winning the race are​ _______.

11.2 7:2

Experiments done without replacement will result in

11.5 Dependent events.

Experiments done with replacement will result in​ ________ events.

11.5 Independent

The formula for finding the probability of event A or event b is P(A or B)=

11.5 P(A) + P(B) - P(A and B).

In how many distinct ways can the letters of the word MEDDLE be​ arranged?

11.7 180 ways

To use an automated teller machine at a certain​ bank, you must enter a 6​-digit code, using the digits 0-9. How many 6​-digit codes are possible if repetition of digits is​ permitted?

11.7 There are 1000000 possible codes.

The number of permutations of n​ objects, where n1 of the items are​ identical, n2 of the items are​ identical, ..., nr are​ identical, is found by

11.7 n! / n1!n2!•••nr!.

How many different committees can be formed from 8 teachers and 39 students if the committee consists of 2 teachers and 3 students?

11.8 8C2 X 39C3 = 255,892

If 5 cards are dealt from a standard deck of​ cards, how many different ways can four face cards and one non-face card be​ dealt? (A face card is a​ king, queen, or​ jack.)

11.8 12C4 X 40C1 = 19,800

If 5 cards are dealt from a standard deck of​ cards, how many different ways can two diamonds and three non-diamonds be​ dealt?

11.8 13C2 X 39C = 712,842

Tina must select and answer in any order four of six essay questions on a test. In how many ways can she do​ so?

11.8 15 ways

If we want to select r items from n​ items, and the order of the arrangement is​ important, then​ _______ are used.

11.8 Permutation

If a sample is drawn in such a way that each time an item is​ selected, each item in the population has an equal chance of being​ drawn, the sample is called a​ _______ sample.

12.1 Random

Beach balls in a basket are shaken and then balls are selected from the basket. Choose the correct sampling technique below.

12.1 Random Sampling

The art and science of​ gathering, analyzing, and making inferences​ (predictions) from numerical information obtained in an experiment is called​ _______.

12.1 Statistics

When a population is divided into​ parts, called​ strata, for the purpose of drawing a​ sample, the procedure is known as a

12.1 Stratified Sampling

Discuss the statement and tell what possible misuse or misinterpretation may exist. At a certain high ​school, half of the students are below average in history. ​Therefore, the school should receive more federal aid to raise student scores.

12.1 The statement is invalid because half of the students in population are expected to be below average in history​, so the school itself is not below average.

Identify the sampling technique used to obtain a sample. The manager at a particular fitness club wants to administer a satisfaction survey to its current members. Using its membership​ roster, the manager randomly selects 200 members and sends them an email with a link to an online survey.

12.1 random Sampling

A line graph with observed values on its horizontal scale and frequencies on its vertical scale is called a frequency

12.2 A frequency polygon

Graphs that are often used to compare parts of one or more components of the whole to the whole are called pie charts or

12.2 Circle Graphs

In a frequency​ distribution, the class with the greatest frequency is call the

12.2 modal class

A number that is representative of a set of data is called​ a(n) _______.

12.3 Average

The Greens are moving. Their real estate agent located 79 houses listed for sale in their price range. Of those houses listed for​ sale, 47 had a finished basement. 54 had a​ three-car garage. 36 had a finished basement and a​ three-car garage.

A)11 B)18 C)65

The Greens are moving. Their real estate agent located 79 houses listed for sale in their price range. Of those houses listed for​ sale, 51 had a finished basement. 58 had a​ three-car garage. 39 had a finished basement and a​ three-car garage.

A)12 B)19 C)70

Circle Graphs

Also known as pie charts are often used to compare parts of one or more components of the whole to the whole

U is the set of colleges in a country. A is the set of colleges that have a baseball team. B is the set of colleges that have a physics department. Describe B ∩ A′ in words.

B ∩ A′ is the set of colleges in a country that have a physics department and do not have a baseball team.

A=​{Tuesday​, Wednesday​, Thursday​, Sunday​} B=​{Tuesday​, Sunday​} Which of the following are true​ statements? Select all that apply.

B ⊆A B ⊂A

3.4 De​ Morgan's laws state that....

De​ Morgan's laws state that ~(p ∧ q) is equivalent to ~p ∨ ~q and ~(p ∨ q) is equivalent to ~p ∧ ~q.

3.3 The conditional statement p → q is only​ _______ when p is true and q is false.

False

If p is true​, q is true​, and r is true​, find the truth value of the statement. (~p ↔ r) ∨ (~q ↔ r)

False

Use a truth table to determine whether the symbolic form of the argument on the right is valid or invalid. ~q↔~r therefore... r→p p→q Choose the correct answer below

False

Determine whether the statement is true or false. If​ false, give the reason. 5 ∉ {1, 3, 5, 7,...} Is the statement true or​ false?

False, because 5 is in the set.

Determine whether the set is finite or infinite. ​{18, 21, 24, ..., 30​}

Finite

Write the negation of the statement. Some vans are on the bridge. Choose the correct answer below.

No vans are on the bridge.

3.1 Write the negation of the statement. Some mopeds are in the crosswalk. Choose the correct answer below.

No mopeds are in the crosswalk.

Systematic Sample

Obtained by selecting a random starting point and then selecting every *nth* item in a population.

8.1 An angle whose measure is greater than 90° but less than 180° is called​ a(n) _______ angle.

Obtuse

Write the negation of the statement. No trucks have three wheels . Choose the correct answer below.

Some trucks have three wheels.

Write the negation of the statement. All turkeys fly. Choose the correct answer below.

Some turkeys do not fly.

3.1 Write the negation of the statement. All pigs fly. Choose the correct answer below.

Some Pigs Do Not Fly

Cluster Sample

Sometimes referred to as an area sample because it is frequently applied on a geographical basis. Essentially, the sampling consists of a random selection of groups of units.

3.4 Conditional statement written as a disjunction

Statements containing connectives other than and & or may have equivalent statements. The conditional statement written as a disjuntion p--> <--> ~p V q

Use a truth table to determine whether the symbolic form of the argument on the right is valid or invalid. r↔q q→p therefore... ~p→~r Choose the correct answer below.

The argument is valid.

Use a truth table to determine whether the symbolic form of the argument on the right is valid or invalid. r↔p p→q therefore... ~q→~r Choose the correct answer below.

The argument is valid.

The Triple L investment club is considering purchasing a certain stock. After considerable​ research, the club members determine that there is a 40​% chance of making ​$12,000​, a 20​% chance of breaking​ even, and a 40​% chance of losing ​$6,000. Find the expectation of this purchase.

The expected value is ​$2400

Use the Venn diagram to list the set A−B′ in roster form.

The letters in between the 2 circles. The shared #

If a club consists of 12 ​members, how many different arrangements of​ president, vice-president, and secretary are​ possible?

The number of possible arrangements is 1320

A set contains eight elements. ​a) How many subsets does it​ have? ​b) How many proper subsets does it​ have?

a) The set has 256 subsets. ​b) The set has 255 proper subsets.

Determine the truth value of the statement (p ∧ q) ∨ [~(p ∧ ~r)] using the following conditions. ​a) p is false​, q is true​, and r is false. ​b) p is true​, q is true​, and r is true

a) True b)True

Evaluate a) 11C3 and b)11P3.

a)165 b)990

3.4 Equivalent

is symbolized <-->( More thicker it has 2 lines going L & R and 2 arrow heads) if both statements have exactly the same truth values in the answer column of the truth tables. Sometimes the words logically equivalent are used in the place of the word equivalent. Another symbol that can be used is the equal sign with an extra line under it so basically 3 lines.

3.5 For the argument​ below, perform the following. ​a) Translate the argument into symbolic form. ​b) Use a truth table to determine whether the argument is valid or invalid. ​(Ignore differences in​ past, present, and future​ tense.) If he's the Scarecrow, he wants a brain. He's not the Scarecrow. therefore

p → q ​~p therefore ... ​~q The argument is invalid because the conclusion is not a tautology.

3.5 The birds are singing but the sun is not out. If the sun is out then the birds are singing. ​Therefore, the sun is out. Write the statement in symbolic form. Is the given argument valid or​ invalid?

p ∧ ~q q → p ... q Invalid

For the argument​ below, perform the following. ​a) Translate the argument into symbolic form. ​b) Determine if the argument is valid or invalid. Compare the argument to a standard form or use a truth table. We will have tacos for dinner, or we will have Chinese food. We will not have tacos for dinner. therefore We will have Chinese food for dinner.

p ∨ q ~p therefore q The argument is valid because it is an example of Disjunctive Syllogism.

8.2 Is the polygon * Square * regular or not​ regular?

regular

Write the statement in symbolic form. Let p and q represent the following statements. ​p: The pizza is delicious. ​q: The cheese is cold. It is false that the pizza is delicious or the cheese is cold.

~(pvq)

Each of the numbers​ 1, 2,​ 3, 4,​ 5, 6,​ 7, 8,​ 9, and 10 are written on a separate piece of paper and placed in a hat. One piece of paper is randomly selected from the hat. Use the Venn diagram shown to help you determine the probability that the number selected is odd or less than 5.

​P(odd or less than 5​)= 7/10

Determine the truth value of the statement (p ∧ q) ∧ (~r ∧ q) using the following conditions. ​a) p is true​, q is false​, and r is true. ​b) p is false​, q is true​, and r is true.

​a) If p is true​, q is false​, and r is true​, what is the value of (p ∧ q) ∧ (~r ∧ q)​? Is FALSE ​b) If p is false​, q is true​, and r is true​, what is the value of (p ∧ q) ∧ (~r ∧ q)​? Is FALSE

3.2 Determine the truth value for each simple statement. Then use these truth values to determine the truth value of the compound statement. 0<−2 or 4≥3

The simple statement 0<−2 is false. The simple statement 4≥3 is true. The compound statement is true.

Determine the truth value for each simple statement. Then use these truth values to determine the truth value of the compound statement. 7+5=17−4 and 50÷5=3•3

The simple statement 7+5=17−4 is false. The simple statement 50÷5=3•3 is false. The compound statement is false.

Determine the truth value for each simple statement. Then use these truth values to determine the truth value of the compound statement. 7+6=18−5 and 28÷7=2•2

The simple statement 7+6=18−5 is true. The simple statement 28÷7=2•2 is true. The compound statement is true.

Determine whether the argument to the right is valid or invalid. You may compare the argument to a standard form or use a truth table. p ↔ q q ∧ r ... p ∨ r Is the argument valid or​ invalid?

Valid

Determine whether the argument to the right is valid or invalid. You may compare the argument to a standard form or use a truth table. p ↔ q r → ~p p ∧ r ... q ∨ ~p Is the argument valid or​ invalid?

Valid

A controlled operation that yields a set of results is called​ a(n) _______.

11.1 experiment

The possible results of an experiment are called its

11.1 outcomes

Probability determined through a study of the possible outcomes that can occur for a given experiment is called

11.1 theoretical probability

A card is picked from a standard deck of 52 cards. Determine the odds against and the odds in favor of selecting an8.

11.2 *Note also that odds against are typically written as a​ ratio, P(failure)​ : P(success).* 1. To determine the odds against selecting an 8​, first determine the probability of selecting an 8. 2.There are four 8s in a standard deck of 52 cards. So 4/52 and simplified is = 1/13 3. Next, determine the probability of not drawing an 8. This can be calculated by subtracting the probability of drawing an 8​, So 1/13​ will get us 12/13. 4. Hence, the odds against drawing an 8 are 12/13 over 1/13. Which Therefore makes the odds against drawing an 8, 12:1. 5. Next, recall that the odds in favor of an event which is success / failure. *Note also that odds in favor are typically written as a​ ratio, P(success)​ : P(failure)* 6.Hence, the odds in favor of drawing an 8 are 1/13 over 12/13. 7.Therefore, the odds in favor of drawing an 8 is 1:12

Tito purchased one raffle ticket as part of a Meals on Wheels fundraiser. The grand​ prize-winning ticket will be randomly drawn from 80 tickets sold. ​a) Determine the probability that Tito wins the grand prize. ​b) Determine the probability that Tito does not win the grand prize. ​c) Determine the odds against Tito winning the grand prize. ​d) Determine the odds in favor of Tito winning the grand prize.

11.2 a) 1/80 b) 79/80 c) 79:1 d) 1:79

A die is tossed. Find the odds against rolling a number greater than 4.

11.2 2:1

The expected gain or loss of an experiment over the long run is called the​ _______ value.

11.3 Expected use the following formula. E=P1•A1+P2•A2+P3•A3+...+Pn•An The variable P1 represents the probability that the first event will​ occur, and A1 represents the net amount won or lost if the first event occurs. P2 is the probability of the second​ event, and A2 is the net amount won or lost if the second event​ occurs, and so on.

A list of all possible outcomes of an experiment is call​ a(n) _______ space.

11.4 Sample

Probability problems that require obtaining a favorable outcome in each of the given events are​ ________ probability problems.

11.5 And

Probability problems that contain the words and or or are considered​ ________ probability problems.

11.5 Compound

For two events A and​ B, if the occurrence of either event in no way affects the probability of the occurrence of the other​ event, then the two events are considered to be​ ________ events.

11.5 Independent

Juan has 3 ties, 5 shirts, and 6 pairs of pants. How many different outfits can he wear if he chooses one​ tie, one​ shirt, and one pair of pants for each​ outfit?

11.7 (6*5*3) 90

At a computer​ store, a customer is considering 10 different​ computers, 8 different​ monitors, 6 different printers and 4 different scanners. Assuming that each of the components is compatible with one another and that one of each is to be​ selected, determine the number of different computer systems possible.

11.7 1920

In a race in which nine automobiles are entered and there are no​ ties, in how many ways can the first three finishers come​ in?

11.7 504vways

The trifecta at most racetracks consists of selecting the​ first-, second-, and​ third-place finishers in a particular race in their proper order. If there are eleven entries in the trifecta​ race, how many tickets must you purchase to guarantee a​ win?

11.7 990 tickets

Which notation expresses the number of permutations of 3 items taken 2 at a​ time?

11.7 The notation nPr represents the number of permutations of n items taken r at a time.

How many permutations are there of the letters in the word ​'CHARGES​', if all the letters are used without​ repetition?

11.7 The number of permutations is 5040.

Identify the formula for the number of permutations when r objects are selected from n objects.

11.7 nPr = n! / (n-r)!

Any ordered arrangement of a given set of objects is called​ a(n)

11.7 permutation

Determine whether the description given describes a permutation or a combination. Jocelyn is the manager of a community college softball team. The team has 16 players. ​a) Jocelyn selects 7 players for the batting order of​ today's game. ​b) Jocelyn selects 4 players to volunteer at the​ school's open house.

11.8 a) Permutation b) Combination

An ice cream store sells 20 flavors of ice cream. Determine the number of 3 dip sundaes.

11.8 1140

A textbook search committee is considering 17 books for possible adoption. The committee has decided to select 4 of the 17 for further consideration. In how many ways can it do​ so?

11.8 2,380

A certain lottery requires players to select 6 different​ numbers, in any​ order, from 1 to 59 inclusive. How many different sets of 6 numbers can be​ chosen?

11.8 59! / 53!X6! = 45,057,474

Some subscribers of a magazine that sells and reviews books respond to an annual questionnaire regarding their satisfaction with the new selection of books. The information obtained from these questionnaires is then used as a sample from which ratings are made by the magazine. Are the data obtained from these returned questionnaires representative of the entire​ population, or are they​ biased? Choose the correct answer below.

12.1 The data is biased because the source of the data comes only from people that subscribe to their magazine.

Discuss the statement and tell what possible misuse or misinterpretation may exist. Company A sells a computer that costs more than the computer sold by Company B. Therefore the Company A computer will last longer than the Company B computer.

12.1 The statement is not valid because more expensive computers are not guaranteed to last longer than less expensive computers.

Discuss the statement and tell what possible misuse or misinterpretation may exist. Suppose eighty-five percent of accidents occur within 12 miles of work. ​Therefore, it is safer not to drive within 12 miles of work.

12.1 The statement is not valid because people may drive within 12 miles of work more often.

When a sample is obtained by drawing every nth​ item, the sample is called a

12.1 systematic sample.

A graph with observed values on its horizontal scale and frequencies on its vertical scale and that uses bars to indicate frequency is called a​ _______.

12.2 histogram.

The average that is found by summing the data and then dividing the sum by the number of pieces of data is called the

12.3 Mean

The value in the middle of a set of ranked data is called the

12.3 Median

The value halfway between the lowest and highest values in a set of data is called the

12.3 Midrange

​a) The symbol for the sample mean is​ _______. ​b) The symbol for the population mean is​ _______.

12.3 a) - X b) weird looking u

The measures of position that divide a set of data into four equal parts are called

12.3 quartiles.

Correlation

12.6 is used to determine whether there is a relationship between two quantities and, if so, how strong the relationship is.

If one quantity increases as the other quantity increases​, the two variables are said to have a​ ________ correlation.

12.6 positive

8.2 The sum of the measures of the interior angles of a triangle is

180 degrees

2.2 The number of distinct proper subsets of a set with n elements is

2^n−1

U is the set of cities in a country. A is the set of cities that have a symphony. B is the set of cities that have a subway system. Describe A ∪ B′ in words. Choose the correct answer below.

A ∪ B′ is the set of cities in a country that have a symphony or do not have a subway system.

Determine the truth value of the statement (~p ∧ q) ∧ (~r ∧ q) using the following conditions. ​a) p is true​, q is true​, and r is true. ​b) p is true​, q is false​, and r is false.

A) False B) False

Jeanine Baker makes floral arrangements. She has 17 different cut flowers and plans to use 6 of them. How many different selections of the 6 flowers are​ possible?

How many ways can the 6 flowers be​ chosen? 12376

If two states are selected at random from a group of 10 ​states, determine the number of possible outcomes if the group of states are selected with replacement or without replacement.

If the states are selected with​ replacement, there are 100 possible outcomes. If the states are selected without​ replacement, there are 90 possible outcomes.

If two states are selected at random from a group of 30 states, determine the number of possible outcomes if the group of states are selected with replacement or without replacement.

If the states are selected with​ replacement, there are 900 possible outcomes. ( 30*30 ) If the states are selected without​ replacement, there are 870 possible outcomes. ( 30 * 29 )

From the 12 male and 12 female sales representatives for an insurance​ company, a team of 2 men and 4 women will be selected to attend a national conference on insurance fraud. In how many ways can the team of 6 be​ selected?

In how many ways can the team of 6 representatives be​ selected? 32670

2.1 A set that is not finite is called​ a(n) _________ set.

Infinite

Determine whether the following set is finite or infinite. A=The set of odd numbers greater than 27. Is the set A finite or​ infinite?

Infinite

Determine whether the set is finite or infinite. The set of odd numbers greater than 5 Which of the following is the correct set​ description?

Infinite

2.2 Determine whether A=B​, A ⊆ B​, B ⊆ A​, A ⊂ B​, B ⊂ A​, or if none of these applies. A=​{8​,7​,6​,5​,4​} B=​{5​,8​,2​,6​,7​}

None of these.

3.1 Write the statement in symbolic form. Let p and q represent the following statements. ​p: The canoes are in the river. ​q: The hotdogs are cooked. The hotdogs are not cooked or the canoes are not in the river. The statement in symbolic form is_______

The statement in symbolic form is ~q∨~p.

3.1 Determine the truth value of each simple statement in the compound statement ​"Florida borders the Atlantic ocean and Alaska borders the Indian ​ocean". Then use these truth values to determine the truth value of the compound statement.

The truth value of the simple​ statement, ​"Florida borders the Atlantic ​ocean." is T. The truth value of the simple​ statement, ​"Alaska borders the Indian ​ocean." is F. The truth value of the compound​ statement, ​"Florida borders the Atlantic ocean and Alaska borders the Indian ​ocean." is F.

If a first experiment can be performed in 7 distinct ways and a second experiment can be performed in 9 distinct​ ways, the two experiments together can be performed in how many distinct​ ways?

There are 63 distinct ways the two experiments can be performed together. (7*9)

Jenny White is shopping for CDs. She decides to purchase 3 movie soundtracks. The music store has 9 different movie soundtracks in stock. How many different selections of movie soundtracks are​ possible?

There are 84 different possible selections of movie soundtracks at the music store.

If p is true​, q is true​, and r is false​, find the truth value of the statement. (p ∧ q) ↔ (q ∨ ~r) Choose the correct answer below.

True because (p ∧ q) is true and (q ∨ ~r) is true.

3.3 If p is true​, q is false​, and r is false​, find the truth value of the statement. (p ∨ q) ↔ (q ∨ ~r) Choose the correct answer below.

True because (p ∨ q) is true and (q ∨ ~r) is true.

A board game uses the deck of 20 cards shown to the right. Two cards are selected at random from this deck. Calculate the probability that both cards selected have a 5​, both with and without replacement

Two cards are to be selected with replacement. Determine the probability that both cards selected have a 5. Is 1/25 Two cards are to be selected without replacement. Determine the probability that both cards selected have a 5. is 3/95

Determine whether the argument to the right is valid or invalid. You may compare the argument to a standard form or use a truth table. p → q q ∧ r r ↔ p ... p Is the argument valid or​ invalid?

Valid

A set contains six elements. ​a) How many subsets does it​ have? ​b) How many proper subsets does it​ have?

a) The set has 64 subsets. ​ ​b) The set has 63 proper subsets.

3.5 If there is an ice storm then the highways are dangerous. If the highways are dangerous then I will not make it to work. Therefore, if there is an ice storm then I will make it to work

. p → q q → ~r ... p → r Invalid

The probability of an event that cannot occur is

11.1 0

Evaluate the expression. ​0!

11.7 1

Simplify. 9​!

11.7 362,880

There is a rack of 15 billiard balls. Balls numbered 1 through 8 are​ solid-colored. Balls numbered 9 through 15 contain stripes. If one ball is selected at​ random, determine the odds for it being striped.

7:8

8.1 Find the measure of the complement of a 5° angle.

85 Right angle is 90 degrees therefore, 90-5=85

Histogram

A Graph with observed values on its horizon scale and frequencies on its vertical scale. A bar is constructed above each observed value ( or class when classes are used ), indicating the frequency of that value ( or class)

Determine the truth value of the statement (p ∧ q) ∧ [~(p ∨ ~r)] using the following conditions. ​a) p is true​, q is false​, and r is true. ​b) p is false​, q is false​, and r is false.

A-False B-False

8.3 A set of points equidistant from a fixed point is called​ a(n) _______.

Circle

3.4 Statements that have exactly the same truth values in the answer columns of their truth tables are called​ __________ statements.

Equivalent

3.2 The disjunction p∨q is false only when both p and q are​ _______.

False

Let p and q represent the following simple statements. ​p: The job pays well. ​q: I get an A. Write the symbolic statement ~(q∨p) in words. Choose the correct sentence below.

It is not true that i get an A or the job pays well

8.1 Two lines in the same plane that do not intersect are called​ _______ lines.

Parallel

A single die is rolled one time. Find the probability of rolling a number greater than 4 or less than 3.

Prob. is 2/3

2.2 Proper Subset ⊂

Set A is a proper subset of set B, symbolized by A ⊂ B, if and only if all the elements of set A are elements of set B and set A (= sign with / through it) set B (that is, set B must contain at least one element not in set A).

2.2 Subset ⊆

Set A is a subset of set B, symbolized by A (c with a line under it looking thing ..... ⊆ ) B, if and only if all elements of set A are also elements of set B.

Write the negation of the statement. All owls fly. Choose the correct answer below.

Some owls do not fly.

3.3 Conditional Statement

The Conditional Statement p--> is true in every case except when p is a true statement and q is a false statement.

Determine whether the argument is valid or invalid. You may compare the argument to a standard form or use a truth table. x → y ~y therefore x Is the argument valid or​ invalid?

The argument is invalid because the conclusion is not a tautology.

Use a truth table to determine whether the symbolic form of the argument on the right is valid or invalid. ~q↔~r therefore... r→p p→q Choose the correct answer below.

The argument is invalid.

Find the supplementary angle of 138.3°.

The supplementary angle of 138.3° is 41.7

variation of the conditional statement

The variations of the conditional statement are the converse of the conditional, the inverse of the conditional, and the contrapositive of the conditional.

3.2 Select the correct choice that completes the sentence below. The conjunction p∧q is true only when both p and q are

True

3.3 The biconditional statement p ↔ q is​ _______ only when p and q have the same truth value.

True

If p is false​, q is true​, and r is true​, find the truth value of the statement. (~p ↔ r) ∨ (~q ↔ r) Select the truth value of (~p ↔ r) ∨ (~q ↔ r) when p is false​, q is true​, and r is true. Choose the correct answer below.

True

3.5 If the conditional statement [(premise 1) ∧ (premise 2)] → conclusion is a​ tautology, then the argument is​ a(n) _______ argument.

Valid

Determine whether the argument is valid or invalid. You may compare the argument to a standard form or use a truth table. m → n ​~n ... ~m Is the argument valid or​ invalid?

Valid

Are M ∩ N and (M' ∪ N')' equal​ statements?

Yes

Are S' ∪ T and (S ∩ T')' equal​ statements?

Yes

3.3 Implication

a conditional statement that is a tautology

A bag contains 9 black chips and 5 white chips. Dexter and Thanh play the following game. Dexter randomly selects one chip from the bag. If the chip is​ black, Dexter gives Thanh ​$4. If the chip is​ white, Thanh gives Dexter ​$13. ​a) Determine​ Dexter's expectation. ​b) Determine​ Thanh's expectation.

a) 2.07 b) -2.07

Determine the truth value of the statement (~p ∧ ~q) ∨ [~(p ∨ r)] using the following conditions. ​a) p is true​, q is false​, and r is true. ​b) p is true​, q is true​, and r is false.

a) False b)False

2.1 Two sets that contain the same elements are called

equal

Let p and q represent the following statements. ​p: The play is boring. ​q: The actors are talented. Write the following statement in its symbolic form. The play is not boring​, but the actors are talented.

he statement ​"The play is not boring​, but the actors are talented​.'' is written ~p∧q.

3.3 Is the statement ~​[(q∨p​)↔~p​] a​ tautology, self-contradiction, or​ neither?

neither

8.1 Two​ angles, the sum of whose measures is 180°​, are called​ _______ angles.

supplementary

3.3 A compound statement that is always true is known as_____

tautology

3.3 Use a truth table to determine whether the statement is a​ tautology, self-contradiction, or neither. p∨​(q∨~p​)

tautology

8.1 When two straight lines​ intersect, the nonadjacent angles formed are called​ _______ angles

vertical

3.1 Let p and q represent the following statements. ​p: The play is boring. ​q: The actors are talented. Write the following statement in its symbolic form. The play is not boring​, but the actors are talented. The statement ​"The play is not boring​, but the actors are talented​.'' is written

is written ~p∧q.

Translate the following statement into symbols. Then construct a truth table for the compound statement and indicate under what conditions the compound statement is true. You did the laundry or you did not leave the garage a mess. ​p: You did the laundry. ​q: You left the garage a mess.

p/ q/p ∨ ~q T TT T FT F TF F FT Use the truth table to determine the set of conditions that make the compound statement true. The statement is true whenever p is true or q is false​, and is false otherwise.

8.3 The sum of the lengths of the sides of a​ two-dimensional figure is called the

perimeter of the figure.

Express the set in roster form. The set of natural numbers between 15 and 153 Choose the correct answer below

{16, 17, 18, 19, ..., 152}

3.4 The inverse of p → q is​ ________.

~p → ~q. The inverse of a logical statement is when the antecedent and consequent remain in the same​ places, but both parts are negated.

Write the statement in symbolic form. Let p and q represent the following statements. ​p: The car is fast. ​q: The tires are large. It is false that the car is fast or the tires are large. Choose the correct answer below.

~​(p∨q)

For the argument​ below, perform the following. ​a) Translate the argument into symbolic form. ​b) Determine if the argument is valid or invalid. Compare the argument to a standard form or use a truth table. If you do a good job, then your boss will be happy. If your boss is happy, then your boss will give you a raise. therefore

.p → q q → r therefore... p → r The argument is valid because it is an example of the Law of Syllogism.

Evaluate the expression 5P3.

11.7 60

To determine the number of distinct ways two or more experiments can be​ performed, the​ _______ principle can be used.

11.7 Counting

The difference between the highest and lowest data values in a data set is called the​ _______.

12.4 range.

The​ symbol, s, is used to indicate the standard deviation of a​ _______.

12.4 sample.

The measure of dispersion that measures how much the data differ from the mean is called the​ _______.

12.4 standard deviation.

By studying the standard deviation​ formula, given​ below, explain why the standard deviation of a set of data will always be greater than or equal to 0. s=Σx−x2n−1 Choose the correct answer below.

12.4 the values of x−x2 will always be greater than or equal to 0.

The value of​ r, the linear correlation​ coefficient, that represents the strongest negative correlation between two variables is

12.6 -1

U is the set of universities in a country. A is the set of universities that have a football team. B is the set of universities that have a business school. Describe B ∩ A′ in words. Choose the correct answer below.

B ∩ A′ is the set of universities in a country that have a business school and do not have a football team.

3.3 Select the truth value of (~p ∧ ~q) ∨ ~r when p is true​, q is true​, and r is true. Choose the correct answer below.

False

2.1 The set of odd numbers greater than 17 Which of the following is the correct set​ description?

Infinite

3.5 Determine whether the argument to the right is valid or invalid. You may compare the argument to a standard form or use a truth table. p → q q ∧ r ... p ↔ r Is the argument valid or​ invalid?

Invalid

An eight​-sided die is tossed. Determine the odds against rolling a 6.

The odds are 7:1

A monk crossbred​ plants, which can have purple or white​ flowers, and obtained 586 plants with white flowers and 234 plants with purple flowers. Find the empirical probability that a plant had each type of flower.

The probability a plant had white flowers is 0.71 ​The probability a plant had purple flowers is .29

A​ single, six-sided die is rolled. Find the probability of rolling an odd number or a number less than 4.

The probability is 2/3

2.2 Decide if the given statement is true or false. If it is​ false, give the reason. 6 ∈ {3,4,6} Choose the correct answer below.

The statement is true​; 6 is an element of {3,4,6}.

2.2 Decide if the given statement is true or false. If it is​ false, give the reason. ​{ ​}=​{∅​} Choose the correct answer below.

The statement is​ false; the set ​{∅​} contains the element ∅.

2.2 Decide if the given statement is true or false. If it is​ false, give the reason. ​{ ​}⊆​{ ​}

The statement is​ true; the empty set is a subset of every​ set, including itself.

Use De​ Morgan's laws to determine whether the two statements are equivalent. ~(~x ∨ ~y), x ∧ y Choose the correct answer below.

The two statements are equivalent

3.4 Use De​ Morgan's laws to determine whether the two statements are equivalent. ~(g ∨ ~h), ~g ∨ h Choose the correct answer below.

The two statements are not equivalent.

Jenny White is shopping for CDs. She decides to purchase 3 movie soundtracks. The music store has 10 different movie soundtracks in stock. How many different selections of movie soundtracks are​ possible?

There are 120 different possible selections of movie soundtracks at the music store.

Five different colored flags will be placed on a​pole, one beneath another. The arrangement of the colors indicates the message. How many messages are possible if five flags are to be selected from fifteen different colored​ flags?

There are 360,360 possible messages.

2.2 Decide if the given statement is true or false. If it is​ false, give the reason. ​{6​,9​,1​} ⊂ {6​,1​,9​}

This statement is false. No set is a proper subset of itself

Tina must select and answer in any order three of ten essay questions on a test. In how many ways can she do​ so?

Tina can select and answer the essay questions on her test in 120 ways.

If p is true​, q is false​, and r is false​, find the truth value of the statement. (~p ∨ ~q) ∨ ~r Select the truth value of (~p ∨ ~q) ∨ ~r when p is true​, q is false​, and r is false. Choose the correct answer below.

True

2.3 Disjoint

Two sets with no elements in common are called disjoint sets.

conveniance sample

Uses data that are easily or readily obtained.

3.5 An argument is​ _______ if the conclusion is true whenever the premises are assumed to be true.

Valid

3.5 Determine whether the argument is valid or invalid. You may compare the argument to a standard form or use a truth table. p ∧ q q ... p Is the argument valid or​ invalid?

Valid

Determine whether the argument to the right is valid or invalid. You may compare the argument to a standard form or use a truth table. p → q r → ~p p ∨ r ... q ∨ ~p Is the argument valid or​ invalid?

Valid

Determine whether the argument to the right is valid or invalid. You may compare the argument to a standard form or use a truth table. p ↔ q q ∧ r ... p → r Is the argument valid or​ invalid?

Valid

Determine the truth value of the statement (p ∧ ~q) ∨ r using the following conditions. ​a) p is false​, q is true​, and r is true. ​b) p is true​, q is false​, and r is true

a) If p is false​, q is true​, and r is true​, what is the truth value of (p ∧ ~q) ∨ r​? TRUE ​b) If p is true​, q is false​, and r is true​, what is the truth value of (p ∧ ~q) ∨ r​? TRUE

2.3 U is the set of cities in a country. A is the set of cities that have a symphony. B is the set of cities that have a science museum. Describe A' in words.

A' is the set of cities in a country that do not have a symphony.

Determine whether the pair of sets is​ equal, equivalent,​ both, or neither. A={geometry, trigonometry, literature} B={trigonometry, literature, geometry} Are set A and set B​ equal, equivalent,​ both, or​ neither?

Both

State whether the statement is true or false. If​ false, give the reason. {p}∈ {f​, d​, x​, s​, p​} Choose the correct answer below.

The statement is false because {t} is a​ set, and not an element of the set.

State whether the following statement is true or false. If​ false, give the reason. A Tale of Two Cities ∈ {books written by Charles Dickens​} Choose the correct answer below.

The statement is true

2.1 Determine whether the following statement is true or false. d∈{t​,d​,j​,q​,m​} Choose the correct answer below.

The statement is true.

2.2 Decide if the given statement is true or false. If it is​ false, give the reason. ​ {guidebook​} ⊆ {newspaper​, instruction manual​, guidebook​} Choose the correct answer below.

The statement is true.

2.2 Decide if the given statement is true or false. If it is​ false, give the reason. {lettuce, beet} ⊆ ​{beet​, lettuce​, tomato​} Choose the correct answer below.

The statement is true​; {lettuce, beet} is a subset of the set ​{beet​, lettuce​, tomato​}

2.2 Decide if the given statement is true or false. If it is​ false, give the reason. 0={ } Choose the correct answer below.

The statement is​ false; 0 is a number and​ { } is a set.

2.1 The three ways a set can be written are​ _______, _______, and​ _______.

The three ways a set can be written are description, roster form, and set-builder notation.

2.2 Determine the number of distinct subsets for the set {S,L,E,D}.

There are 4 #'s of distinct subsets, Therefore plug it into the equation 2^n. 2^4 = 16

2.2 Decide if the given statement is true or false. If it is​ false, give the reason. Game Box ∈ {Port'n'Play​, Game Box​, Funsole​}

This statement is true.

2.5 The Greens are moving. Their real estate agent located 77 houses listed for sale in their price range. Of those houses listed for​ sale, 49 had a finished basement. 54 had a​ three-car garage. 38 had a finished basement and a​ three-car garage.

a) How many had a finished basement but not a​ three-car garage? 49-38=11 ​b) How many had a​ three-car garage but not a finished​ basement? 54-38=16 ​c) How many had either a finished basement or a​ three-car garage? 11+16+38=65

Identify the sampling technique used to obtain a sample. A famous magazine chooses its ​"funniest athletes​" by compiling responses from readers who mail in a survey printed in the magazine.

12.1 Convenience sampling

The​ collection, organization, and analysis of data is called​ _______ statistics.

12.1 Descriptive

A group of people are classified according to food preference and then random samples of people from each group are taken. Choose the correct sampling technique below.

12.1 Stratified sampling

Identify the sampling technique used. Every 7th person in line to buy tickets to a concert is asked his or her age.

12.1 Systematic

Discuss the statement and tell what possible misuse or misinterpretation may exist. At a certain middle ​school, half of the students are below average in social studies. ​Therefore, the school should receive more federal aid to raise student scores.

12.1 The statement is invalid because half of the students in population are expected to be below average in social studies​, so the school itself is not below average.

A subset of a population used by statisticians to make predictions about a population is called a​ _______.

12.1 sample

8.3 The region within the boundaries of a​ two-dimensional figure is called the

area of the figure.

2.2 The expression for determining the number of distinct subsets for a set with n distinct elements is​ ______.

2^1

2.1 Finite Set

A set that contains no elements or the number of elements in the set is a natural number is called a finite set.

3.1 Write the negation of the statement. Some birds do not have claws. Choose the correct answer below.

All Birds Have Claws

2.1 Express the following set in​ set-builder notation. B={9, 10, 11, 12, 13, 14, 15, 16} Choose the correct answer below.

B=​{x| x∈N and 9≤x≤16​}

2.1 The number of elements in a set is called the ________ number of the set.

Cardinal

2.1 Determine whether the set is finite or infinite. ​{35, 40, 45, 50, ...​}

Infinite

2.2 Number of distinct Proper Subsets

The number of distinct proper subsets of a finite set A is 2^n-1,where n is the # of elements in set A.

2.2 Number of distinct subsets

The number of distinct subsets of a finite set A is 2^n , where (n) is the number of elements in set A.

2.3 Intersection

The set containing all the elements that are common to both set A and set B is called the intersection of set A and set B.

2.2 Decide if the given statement is true or false. If it is​ false, give the reason. {lettuce, beet}⊆{beet​, lettuce​, tomato​}

The statement is true; {lettuce, beet} is a subset of the set {beet​, lettuce​, tomato​}

2.2 Decide if the given statement is true or false. If it is​ false, give the reason. ​{ ​} ⊆ {beet​, lettuce​, tomato​}

The statement is​ true; { ​} is a subset of the set.

2.2 Decide if the given statement is true or false. If it is​ false, give the reason. ​{9​} ∈ {{7​}, {8​}, {9​}} Choose the correct answer below.

The statement is​ true; {9​} is an element of the set.

2.2 Decide if the given statement is true or false. If it is​ false, give the reason. ​{ferry​, row boat​} ⊂ ​{row boat​, fishing boat​, ferry​, kayak​}

This statement is true.

2.1 Ellipsis

Three dots placed in a set to show that the set continues in the same manner is called an ellipsis.

2.2 List all the subsets of the given set. ​{nectarine​}, {pear​}, {lemon​}

{}, ​{nectarine​}, ​{pear​}, {lemon​}, ​{nectarine​, pear​}, ​{nectarine​, lemon​}, ​{pear​,lemon​}, ​{nectarine​, pear​,lemon​}

The last 60 boat rentals at a boat rental company were 38 sunfish, 10 kayaks, and 12 rowboats. a) Determine the empirical probability that the next boat rental is a sunfish. ​b) Determine the empirical probability that the next boat rental is a kayak. ​c) Determine the empirical probability that the next boat rental is a rowboat.

11.1 a) 19/30 b) 1/6 c) 1/5

Every probability must be a number between

11.1 0 and 1

The probability of an event that must occur is

11.1 1

8.2 Is the polygon * Triangle * regular or not​ regular? Diff lengths

Not Regular

If we leave at 9:00, then we will get there in time. We will get there in time. therefore We left at 9:00.

p → q q therefore p The argument is invalid because the argument is an example of the Fallacy of the Converse.

Let p and q represent the following statements. ​p: I get an A. ​q: I graduate. Write the following compound statement in its symbolic form. I do not graduate if and only if I get an A. The statement ​"I do not graduate if and only if I get an A​.'' is written

~q↔p.

range

12.4 The diff between the highest and lowest values. It indicates the total spread of the data. The rage of a set of data can be calculated using the formula Range=highest value - Lowest value.

A set of points representing data is called​ a/an __________ diagram.

12.6 Scatter

3.3 biconditional statement

A statement that contains the phrase "if and only if" p<---> is true only when p & q have the same truth value, that is, when both are true or both are false.

Determine the truth value of the statement (~r ∧ p) ∨ q using the following conditions. ​a) p is false​, q is true​, and r is true. ​b) p is true​, q is false​, and r is true.

A) True B) False

Determine the truth value of the statement (~r ∨ ~p) ∧ q using the following conditions. ​a) p is true​, q is false​, and r is false. ​b) p is true​, q is true​, and r is true.

A)False B)False

According to a disease control​ center, 37​% of people in a certain country have high blood pressure. If a person in this country is selected at​ random, determine the odds in favor of this person having high blood pressure.

11.2 The odds are 37:63.

Suppose that the probability that all the parts needed to assemble a bookcase are included in the carton is 4/5. Determine the odds in favor of the carton including all the needed parts.

11.2 The odds are 4:1

The ratio of the probability that the event will occur to the probability that the event will fail to occur is called the odds

11.2 in favor of an event

A listing of observed values and the corresponding frequency of occurrence of each value is called a​ _______ distribution.

12.2 frequency distribution.

3.5 For the argument​ below, perform the following. ​a) Translate the argument into symbolic form. ​b) Use a truth table to determine whether the argument is valid or invalid. ​(Ignore differences in​ past, present, and future​ tense.) If he's the Tin Man, he wants a heart. He's the Tin Man. therefore

p → q p therefore... q The argument is valid because the conclusion is a tautology.

3.1 ​Statement: If it is sunny, then Marvin will go for a run. A)Is the above statement simple or​ compound?

A) The statement is a compound statement because it combines two or more simple statements. B)The statement is a conditional. It is represented by the symbol then →.

2.3 U is the set of colleges in a country. A is the set of colleges that have a football team. B is the set of colleges that have a business school. Describe A ∩ B in words.

A ∩ B is the set of colleges in a country that have both a football team and a business school.

2.2 Determine whether A=​B, A ⊆ ​B, B ⊆ ​A, A ⊂ ​B, B ⊂ ​A, or if none of these applies. Set A is the set of people living in South Carolina. Set B is the set of people living in a state with a border on the Atlantic Ocean.

A ⊂ B

2.2 Decide if the given statement is true or false. If it is​ false, give the reason. ​{f​, e​, g​} ⊄ {e​, f​, g​} Choose the correct answer below.

The statement is​ true; ​{f​,e​,g​} is a not a proper subset of the set.

Thirty-six cities were researched to determine whether they had a professional sports​team, a​symphony, or a ​children's museum. Of these​ cities, 15 had a professional sports​team. 16 had a​symphony. 15 had a​children's museum. 8 had a professional sports team and a​symphony. 5 had a professional sports team and a​children's museum. 8 had a symphony and a​children's museum. 3 had all three activities.

a) How many of the cities surveyed had only a professional sports​ team? 8-3=5 ​b) How many of the cities surveyed had a professional sports team and a​ symphony, but not a​ children's museum? 8-3=5 c) How many of the cities surveyed had a professional sports team or a​ symphony?

Write the statement in symbolic form. Let p and q represent the following statements. ​p: The radio is loud. ​q: The clock is small. It is false that the radio is loud or the clock is small. Choose the correct answer below.

. ​~(pvq)

Evaluate the expression 9P9.

11.7 362880

Random Sample

A sample that is drawn in such a way that each time an item is selected each item in the population has an equal chance of being drawn

A=​{winter​, summer​, spring​, autumn​} B=​{winter​, autumn​} Which of the following are true​ statements? Select all that apply.

B⊂A B ⊆ A

3.3 Tautology

Compound statement that is always true

To determine the validity of an argument with two​ premises, construct a truth table for a conditional statement of the form [(premise 1) ∧ (premise 2)] →

Conclusion

Express the set in roster form. B={x | x∈N and x≥3} Choose the correct answer below.

D. B= { 3,4,5,6,...}

Write the negation of the statement. Some mopeds are in the parking lot. Choose the correct answer below.

No mopeds are in the parking lot.

Write the negation of the statement. Some scooters are in the parking lot. Choose the correct answer below.

No scooters are in the parking lot.

8.2 Is the polygon * Trapezoid * regular or not​ regular?

Not Reg

3.5 Determine whether the argument is valid or invalid. You may compare the argument to a standard form or use a truth table. u → v ~v therefore u Is the argument valid or​ invalid?

The argument is invalid because the conclusion is not a tautology.

Use a truth table to determine whether the symbolic form of the argument on the right is valid or invalid. ~r↔~p p→q therefore... q→r Choose the correct answer below.

The argument is invalid.

3.5 Determine whether the argument is valid or invalid. You may compare the argument to a standard form or use a truth table. ~q→~p r→q ...p→q Choose the correct answer below.

The argument is valid because the conclusion is a tautology.

A card is picked from a standard deck of 52 cards. Determine the odds against and the odds in favor of selecting a queen.

The odds against selecting a queen are 12:1 The odds in favor of selecting a queen are 1:12

If the probability that an event occurs is 0.5​, the probability that the event does not occur is​ _________.

The probability the event does not occur is . 5.

If p is true​, q is false​, and r is false​, find the truth value of the statement. (~p ∨ ~q) ∨ ~r

True

3.2 The only way a conjunction ( p ^ q ) is true is when

both p & q are true

3.5 An argument that is invalid is also known as a​ _______.

fallacy

3.4 The converse of p→q is​ _______.

p→q is q→p .

Procedure: Rules for Data Grouped By Classes

1. The classes should be the same "Width." 2.The Classes Should Not Overlap. 3. Each Piece Of Data Should Belong To Only One Class.

The sum of the probabilities of all possible outcomes of an experiment is​ _______.

11.1 1

One card is selected at random from a deck of cards. Determine the probability that the card selected is a 9. ( 4 of each # )

11.1 1/13

One card is drawn from a standard 52​-card deck. Determine the probability that the card selected is a red card. (26 cards are red in a deck)

11.1 1/2

You are dealt one card from a standard​ 52-card deck. Find the probability of being dealt a heart. The probability of being dealt a heart is (13 of each suite)

11.1 1/4

You are dealt one card from a standard​ 52-card deck. Find the probability of being dealt the ace of spades.

11.1 1/52

One card is selected at random from a deck of cards. Determine the probability that the card selected is not a 3.

11.1 12/13

One card is drawn from a standard 52​-card deck. Determine the probability that the card selected is not a diamond.

11.1 3/4

If each outcome of an experiment has the same chance of occurring as any other​ outcome, the outcomes are​ _______ likely outcomes.

11.1 equally

A subcollection of the outcomes of an experiment is called​ a(n) _______.

11.1 event

If the odds against an event are 1​:4​, then the probability that the event will fail to occur is​ ________. (denominator is always the 2 numbers added)

11.2 1/5

If the probability that an event will occur is 1/3​, then the probability that the event will not occur is 2/3​, and the odds in favor of the event occurring are​ ________.

11.2 1:2

A fair six​-sided die is tossed. Determine the odds against rolling a number less than 2.

11.2 5:1

A die is tossed. Find the odds against rolling a number greater than 1.

11.2 The odds against rolling a number greater than 1 are 11​:55.

A card is picked from a standard deck of 52 cards. Determine the odds against and the odds in favor of selecting a heart.

11.2 The odds against selecting a heart are 3:1 The odds in favor of selecting a heart are 1:3

The ratio of the probability that an event will fail to occur to the probability that the event will occur is called the odds​ _______ an event.

11.2 against

In an​ experiment, if there is a loss in the long​ run, the expected value is

11.3 Negative

In an​ experiment, an individual expects to have a gain in the long​ run, the expected value is​ _______.

11.3 Positive

In an​ experiment, if there is neither a gain nor a loss in the long​ run, the expected value is

11.3 Zero 0

Each individual outcome in a sample space is called a sample

11.4 point

If a club consists of 16 members, how many different arrangements of​ president, vice-president, and secretary are​ possible?

11.7 3360

How many tickets with different points of origin and destination can be sold on a bus line that travels a loop with 28 stops?

11.7 756 diff tix can be sold.

In one question on a history​ test, a student is asked to match 7 dates with 7 events; each date can be matched with only 1 event. In how many ways can this question be​ answered?

11.7 This question can be answered in 5040 different ways.

The formula for the number of permutations of n distinct items is n!

11.7 ​n!=n(n−1)(n−2)•••(3)(2)(1).

A quinella bet consists of selecting the​ first- and​ second-place winners, in any​ order, in a particular event. For​ example, suppose you select a​ 2-5 quinella. If 2 wins and 5 finishes​ second, or if 5 wins and 2 finishes​ second, you win. Mr. Smith goes to a jai alai match. In the​ match, 7 jai alai teams compete. How many quinella tickets must Mr. Smith purchase to guarantee a​ win?

11.8 7! / (7-1=5)! X 2! = 21 quinella tickets must be purchased to guarantee a win.

A distinct group of objects without regard to their arrangement is called a​ _______.

11.8 Combination

Making generalizations or predictions from the data collected is called

12.1 Inferential statistics

Discuss the statement and tell what possible misuse or misinterpretation may exist. Eighty-five percent of all dentists recommend Brand A mouthwash. ​Therefore, this mouthwash is better than all other types of mouthwash.

12.1 The statement is not valid because a recommended mouthwash may not be better than other types of mouthwash.

Discuss the statement and tell what possible misuse or misinterpretation may exist. There are more empty spaces in the parking lot of a Japanese restaurant than at a Mexican restaurant.​ Therefore, more people prefer Mexican food than Japanese food.

12.1 The statement is not valid because more people may walk to the Japanese restaurant than the Mexican restaurant.

In a study of patients with flu symptoms​, each patient was found to have improved symptoms after taking echinacea. Therefore, echinacea cures the flu. Discuss the statement and tell what possible misuse or misinterpretation may exist. Choose the correct answer below.

12.1 The statement is not valid because the patients may have improved on their own without taking echinacea and having improved symptoms does not necessarily translate to the patients being cured of the flu.

Discuss the statement and tell what possible misuse or misinterpretation may exist. The average thickness of the ice is 7 in​, so the pond is frozen thick enough to skate on.

12.1 The statement is not valid because there may be sections of thin ice​, so it may not be frozen thick enough to skate on.

Discuss the statement and tell what possible misuse or misinterpretation may exist. Thirty students said that they would recommend Professor Lowenthal to a friend. Twenty students said that they would recommend Professor Wagner to a friend.​ Therefore, Professor Lowenthal is a better teacher than Professor Wagner.

12.1 The statement is not valid because we​ don't know how many of each​ professor's students were surveyed and student opinion does not necessarily determine the quality of a​ professor's teaching abilities.

A sample that consists of a random selection of groups or units is called a​ _______ sample.

12.1 cluster

All items or people of interest in an experiment are collectively called a

12.1 population.

Fill in the blank with the appropriate value. The class width of a frequency distribution with a first class of 10-19 and a second class of 20-29 is

12.2 10

In a frequency​ distribution, another name for the midpoint of a class is the class​ _______.

12.2 Mark

In a​ stem-and-leaf display, ​a) the left group of digits is called the b) the right group of digits is called the

12.2 a) Stem b)Leaf

The piece of data that occurs most frequently is called the​ _______.

12.3 Mode

The symbol σ is used to indicate the standard deviation of a

12.4 Population

Measures of dispersion are used to indicate the spread or

12.4 variability of the data.

Without actually doing the​ calculations, decide​ which, if​ either, of the following two sets of data will have the greater standard deviation. Explain why. 11​,14​,15​,16​,18​,22 15​,16​,16​,17​,17​,

12.4 The first set will have the greater standard deviation because the data have a greater spread about the mean.

The value of​ r, the linear correlation​ coefficient, that represents no correlation between two variables is

12.6 0

The value of​ r, the linear correlation​ coefficient, that represents the strongest positive correlation between two variables is

12.6 1

A unitless measure that describes the strength of the linear relationship between two variables is called the linear correlation​ __________.

12.6 coefficient, r

A set contains twelve elements. ​a) How many subsets does it​ have? ​b) How many proper subsets does it​ have?

2^12 power ​a) The set has 4096 subsets. ​ ​b) The set has 4095 proper subsets.

2.2 cont... How many of the distinct subsets are proper subsets?

2^n - 1 {S,L,E,D} has 4 elements in the set 2^4 = 16(-1) =15 are proper subsets The one that gets eliminated is {S,L,E,D} because it can not match 100%.

Evaluate the expression. 7C4

7C4 = 7!/ 4! X 3! = 35

2.4 The number of regions created when constructing a Venn diagram with three overlapping sets is​ _______.

8

3.4 De Morgan's Law

A set of rules for converting an expression containing NOTs into an expression that does not contain any NOTs. 1.~(p^q) <--> ~p V ~q 2.~(pVq) <--> ~p ^ ~q

2.3 U is the set of cities in a country. A is the set of cities that have a zoo. B is the set of cities that have a symphony. Describe A ∩ B′ in words.

A ∩ B′ is the set of cities in a country that have a zoo and do not have a symphony.

2.3 For the sets​ U, A, and​ B, construct a Venn diagram and place the elements in the proper regions. U=​{1,2, 3,​ 4, 5,​ 6, 7,​ 8} A={1, 3, 4} B={3, 4, 5, 6, 7}

A(Has 1 in its circle) B(Has 6,7,5 in it's circle) The middle has (8,2) On the outside of the venn Diagram is (3,4)

In a survey of employees at a fast food​ restaurant, it was determined that 13 cooked​ food, 15 washed​ dishes, 25 operated the cash​ register, 8 cooked food and washed​ dishes, 6 cooked food and operated the cash​ register, 8 washed dishes and operated the cash​ register, 3 did all three​ jobs, and 5 did none of these jobs

A) 39 B) 2 C) 14 D) 5 E) 21 F) 16

3.1 Select the correct choices that complete the sentences below. ​a) The conditional is symbolized by → and is read​ "_______." ​b) The biconditional is symbolized by ↔ and is read​ "_______."

A) If-Then B) If and Only If

Determine the truth value of the statement (r ∨ p) ∨ ~q using the following conditions. ​a) p is false​, q is false​, and r is true. ​b) p is false​, q is true​, and r is false.

A) True B) False

In a survey of employees at a fast food​ restaurant, it was determined that 13 cooked​ food, 14 washed​ dishes, 15 operated the cash​ register, 7 cooked food and washed​ dishes, 3 cooked food and operated the cash​ register, 4 washed dishes and operated the cash​ register, 2 did all three​ jobs, and 5 did none of these jobs.

A)35 B)5 C)10 D)2 E)17 F)10 (include middle #)

8.1 The union of two rays with a common endpoint is called​ a(n) _______.

Angle

3.3 A compound statement that is always false is known as a​ _______.

A​ self-contradiction is a compound statement that is always false. When every truth value in the answer column of the truth table is​ false, then the statement is a​ self-contradiction.

U is the set of universities in a country. A is the set of universities that have a baseball team. B is the set of universities that have a humanities program. Describe B ∩ A′ in words. Choose the correct answer below.

B ∩ A′ is the set of universities in a country that have a humanities program and do not have a baseball team.

U is the set of universities in a country. A is the set of universities that have a football team. B is the set of universities that have a business school. Describe B ∪ A′ in words. Choose the correct answer below.

B ∪ A′ is the set of universities in a country that have a business school or do not have a football team.

Determine whether A=​B, A ⊆ ​B, B ⊆ ​A, A ⊂ ​B, B ⊂ ​A, or if none of these applies. A=​{spring​, winter​, autumn​, summer​} B=​{spring​, summer​} Which of the following are true​ statements? Select all that apply.

B ⊂ A & B ⊆ A

2.2 Determine whether A=​B, A ⊆ ​B, B ⊆ ​A, A ⊂ ​B, B ⊂ ​A, or if none of these applies. A=​{spring​, winter​, autumn​, summer​} B=​{spring​, summer​}

B ⊂ ​A

2.2 Determine whether A=​B, A ⊆ ​B, B ⊆ ​A, A ⊂ ​B, B ⊂ ​A, or if none of these applies. A=​{x|x is a sport that uses a ball​} B=​{jai alai​, field hockey​, tennis​,soccer​}

B ⊆ A B ⊂A

Determine the truth value of the statement (r ∨ ~p) ∨ q using the following conditions. ​a) p is false​, q is false​, and r is false. ​b) p is true​, q is true​, and r is false.

Both True

A state is divided into regions using towns. A random sample of 30 town areas is selected. Choose the correct sampling technique below.

Cluster sampling

Identify the sampling technique used to obtain the following sample. A state is divided into regions using counties. A random sample of 20 county areas is selected. Choose the correct sampling technique below.

Cluster sampling

3.1 A statement that consists of two or more simple statements is called a

Compound Statement

2.3 U is the set of office supply stores. A is the set of office supply stores that sell markers. B is the set of office supply stores that sell chairs. C is the set of office supply stores that sell binders. Describe C′ ∩ B.

C′ ∩ B is the set of office supply stores that sell chairs and do not sell binders

3.5 For the argument​ below, perform the following. ​a) Translate the argument into symbolic form. ​b) Determine if the argument is valid or invalid. Compare the argument to a standard form or use a truth table. You will do what I say, or you will be punished. You will not do what I say. therefore You will be punished.

D. p ∨ q ~p therefore... q The argument is valid because it is an example of Disjunctive Syllogism.

2.1 The set that contains no elements is called the empty or null set.

Empty or null

2.1 Two sets that contain the same number of elements are called​ _______.

Equivalent

Of the​ converse, inverse, and​ contrapositive, only the contrapositive of the conditional statement is ▼ to the conditional statement.

Equivalent.. Only the contrapositive of the conditional statement is equivalent to the conditional statement.

3.3 Select the truth value of a → (b → c) when a is true​, b is true​, and c is false. Choose the correct answer below.

False

If p is true​, q is true​, and r is true​, find the truth value of the statement. (~p ↔ r) ∧ (~q ↔ r) Select the truth value of (~p ↔ r) ∧ (~q ↔ r) when p is true​, q is true​, and r is true. Choose the correct answer below.

False

2.3 Venn Diagram

In a Venn Diagram, a rectangle usually represents the universal set, U. The items inside the rectangle may be divided into subsets of the universal set.

How many different committees can be formed from 10 teachers and 42 students if the committee consists of 3 teachers and 3 ​students?

In how many ways can the committee of 6 members be​ selected? 1377600

From the 11 male and 10 female sales representatives for an insurance​ company, a team of 4 men and 2 women will be selected to attend a national conference on insurance fraud. In how many ways can the team of 6 be​ selected?

In how many ways can the team of 6 representatives be​ selected? 14850

2.1 Determine whether the following set is finite or infinite. A= The set of multiples of 5 greater than 55

Infinitate

Determine whether the set is finite or infinite. ​{56, 64, 72, 80, ...​} Choose the correct answer below.

Infinite

3.5 When the conclusion of an argument does not necessarily follow from the given set of premises it is​ a(n) ------- argument

Invalid

Determine whether the argument to the right is valid or invalid. You may compare the argument to a standard form or use a truth table. p → q q ∨ r ... p ↔ r Is the argument valid or​ invalid?

Invalid

Determine whether the argument to the right is valid or invalid. You may compare the argument to a standard form or use a truth table. p ↔ q q ∨ r ... p ∧ r Is the argument valid or​ invalid?

Invalid

Determine whether the argument to the right is valid or invalid. You may compare the argument to a standard form or use a truth table. p ∨ q r ∧ p ... q Is the argument valid or​ invalid?

Invalid

If the conditional statement of the form [(premise 1) ∧ (premise 2)] → conclusion is not a ​tautology, then the argument is​ a(n) _______ argument.

Invalid

Stratified Sampling

Involves dividing the population by characteristics called stratifying factors, such as gender, race, and religion, or income.

3.3 Self-Contradiction

Is a compound statement that is always FALSE.

Frequency Distribution

Is a listing of the observed values and the corresponding frequency of occurrence of each value.

3.5 Draw a valid conclusion from the given premises. If my son is afraid of the dark, then he needs a nightlight. My son is afraid of the dark. ​Therefore, ...

My son needs a nightlight.

Use the Venn diagram to determine whether the statements are equal for all sets A and B. A′ ∪ B and A′ ∩ B

No

Use the Venn diagram to determine whether the statements are equal for all sets A and B. A′ ∪ B′ and A′ ∩ B

No

Use the Venn diagram to determine whether the statements are equal for all sets A and B. A ∪ B and A′ ∩ B′

No

8.1 An angle that measures 90° is​ a(n) _____ angle.

Right

Write the negation of the statement. All parrots fly. Choose the correct answer below.

Some parrots do not fly.

Write the negation of the statement. All pigs fly. Choose the correct answer below.

Some pigs do not fly.

3.1 A sentence that can be judged either true or false is called​ a(n)

Statement

Identify the sampling technique used to obtain the following sample. A group of people are classified according to race and then random samples of people from each group are taken.

Stratified sampling

2.2 If all the elements of set A are also elements of set​ B, then set A is​ a(n) _______ of set B

Subset

A survey of 600 farmers showed that of the​ farmers, 134 grew only​ wheat, 121 grew only​ corn, 103 grew only​ oats, 252 grew​ wheat, 84 grew wheat and​ corn, 82 grew wheat and​ oats 234 grew corn. Determine the number of farmers who ​a) grew at least one of the​ three, ​b) grew all​ three, ​c) did not grow any of the​ three, and ​d) grew exactly two of the three.

Let A represent the farmers that grew wheat. Let B represent the farmers that grew corn. Let C represent the farmers that grew oats. The Venn diagram is shown to the right. Note that 134 grew only wheat. This group is represented on the Venn diagram by region I. ​Next, note that 121 grew only corn. This group is represented on the Venn diagram by region III. The problem states that 103 grew only oats. This group is represented on the Venn diagram by region VII. This is shown on the diagram to the right. The problem states that 252 grew wheat. This group is represented on the Venn diagram by regions​ I, II,​ IV, and V. Since region I is known to contain 134​, regions​ II, IV, and V must contain a total of 252−134​, or 118. The problem states that 84 grew wheat and corn. This group is represented on the Venn diagram by regions II and V. Note that regions​ II, IV, and V contain a total of 118​, and regions II and V contain a total of 84. To find the number in region​ IV, subtract 84 from 118. Therefore, region IV must contain 118−84​, or 34. The problem states that 82 grew wheat and oats. This group is represented on the Venn diagram by regions IV and V. Since region IV is known to contain 34​, region V must contain 82−34​, or 48. Recall that 84 grew wheat and​ corn, which is represented by regions II and V. Since region V is now known to contain 48​, region II must contain 84−48​, or 36. The problem states that 234 grew corn. This group is represented on the Venn diagram by regions​ II, III,​ V, and VI. Note that region II is known to contain 36​, region III is known to contain 121​, and region V is known to contain 48. Therefore, region VI must contain 234−36−121−48​, or 29. To complete the Venn​ diagram, only the number in region VIII must be found. The problem states that 600 farmers were surveyed. The number in each of the regions I through VII is known.​ Therefore, region VIII must contain 600−134−36−121−34−48−29−103​, or 95. The group of farmers who grew at least one of the three crops is represented by regions​ I, II,​ III, IV,​ V, VI, and VII. Find the sum of the numbers in each region. 134+36+121+34+48+29+103=505 ​Therefore, 505 farmers grew at least one of the three crops. ​b) The group of farmers that grew all three crops is represented on the Venn diagram by region V. On the Venn​ diagram, region V contains 48 farmers. ​Therefore, 48 farmers grew all three crops. c) The group of farmers that grew none of the three crops is represented on the Venn diagram by region VIII. On the Venn​ diagram, region VIII contains 95 farmers. ​Therefore, 95 farmers grew none of the three crops. d) The group of farmers that grew exactly two of the three crops is represented on the Venn diagram by regions​ II, IV, and VI. Find the sum of the numbers in each region. 36+34+29=99 ​Therefore, 99 farmers grew exactly two of the three crops.

The following statistics represent weekly salaries at a construction company. Mean ​$590 First quartile ​$495 Median ​$555 Third quartile ​$665 Mode ​$630 87th percentile ​$765

The most common salary is ​$630630. The salary that half the​ employees' salaries surpass is ​$555555. The percent of​ employees' salaries that surpassed ​$665 is 2525​%. The percent of​ employees' salaries that were less than ​$495 is 2525​%. The percent of​ employees' salaries that surpassed ​$765 is 1313​%. If the company has 100 ​employees, the total weekly salary of all employees is ​$5900059000.

There is a rack of 15 billiard balls. Balls numbered 1 through 8 are​ solid-colored. Balls numbered 9 through 15 contain stripes. If one ball is selected at​ random, determine the odds against it being solid-colored.

The odds are 7:8

Casey is going to wear a gray sportcoat and is trying to decide what tie he should wear to work. In his​ closet, he has 42 ​ties, 21 of which he feels go well with the sportcoat. If Casey selects one tie at​ random, determine the probablity and the odds of the tie going well or not going well with the sportcoat.

The probability the tie goes well with the jacket is 1/2 The probability the tie will not go well with the jacket is 1/2 The odds against the tie going well with the jacket is 1:1 The odds in favor of the tie going well with the jacket is 1:1

Find the range and standard deviation of the set of data. 3​, 7​, 3​, 7​, 7​, 10​, 12

The range is 9 The standard deviation is 3.32

2.3 Union

The set containing all the elements that are members of set A or set B or of both is called the union of set A and set B.

2.2 A set contains eight elements. ​a) How many subsets does it​ have? ​b) How many proper subsets does it​ have?

The set has 256 subsets. ​b) The set has 255 proper subsets. 2^8 power

U is the set of furniture stores. A is the set of furniture stores that sell mattresses. B is the set of furniture stores that sell cabinets. C is the set of furniture stores that sell leather furniture. Describe the set A ∪ B ∪ C in words

The set of furniture stores that sell mattresses or cabinets or leather

3.3 Determine whether the statement q → ~q is an implication

The statement is not an implication.

Let p and q represent the following statements. ​p: I work hard. ​q: I get a raise. Write the following compound statement in its symbolic form. I do not get a raise if and only if I work hard.

The statement ​"I do not get a raise if and only if I work hard​.'' is written ~q↔p.

Let p and q represent the following statements. ​p: I eat bananas. ​q: I like ice cream. Write the following compound statement in its symbolic form. I do not like ice cream if and only if I eat bananas.

The statement ​"I do not like ice cream if and only if I eat bananas​.'' is written ~q↔p.

Let p and q represent the following statements. ​p: The job pays well. ​q: The taxes are high. Write the following compound statement in its symbolic form. The taxes are not high if and only if the job pays well. The statement ​"The taxes are not high if and only if the job pays well​.'' is written

The statement ​"The taxes are not high if and only if the job pays well​.'' is written ~q↔p.

Discuss the statement and tell what possible misuse or misinterpretation may exist. Suppose ninety percent of accidents occur within 15 miles of home. ​Therefore, it is safer not to drive within 15 miles of home.

The statement is not valid because people may drive within 15 miles of home more often.

3.4 The conditional statement p → q is equivalent to the following disjunction​ statement: _________.

The statements p → q and​ ~p ∨ q are equivalent since they have exactly the same truth values in the answer columns of their truth tables.

3.1 Let p and q represent the following simple​ statements: ​p: The play is boring. ​q: The taxes are high. Write the following compound statement in its symbolic form. The play is boring and the taxes are high. The symbolic form is

The symbolic form is p∧q.

Let p and q represent the following simple​ statements: ​p: The stove is hot. ​q: I study. Write the following compound statement in its symbolic form. The stove is hot and I study.

The symbolic form is p∧q.

Let p and q represent the following statements. ​p: The tie is red. ​q: The scarf is grey. Write the compound statement p ∨ q in words.

The tie is red or the scarf is grey

Use De​ Morgan's laws to determine whether the two statements are equivalent. ~(p ∧ q), ~p ∧ q Choose the correct answer below.

The two statements are not equivalent.

2.2 Decide if the given statement is true or false. If it is​ false, give the reason. ​{green​} ⊂ {green​, blue​, brown​}

This statement is true.

2.2 Decide if the given statement is true or false. If it is​ false, give the reason. ​{guidebook​} ⊆ ​{newspaper​, instruction manual​, guidebook​} Choose the correct answer below.

This statement is true.

If p is true​, q is true​, and r is false​, find the truth value of the statement. (~p ∧ ~q) ∨ ~r Select the truth value of (~p ∧ ~q) ∨ ~r when p is true​, q is true​, and r is false. Choose the correct answer below.

True

8.1 Find the complementary angle of 40 4/5°.

Two angles are called complementary angles if the sum of their measures is​ 90°. Set up an equation using the known angle and the relationship described above. Let x represent the measure of the complementary angle. 40 4/5 + X = 90 Subtract 40 4/5 from each sides of the equation. Therefore, 90 - 45 4/5 = 49 1/5. Thus, the complementary angle is 49 1/5.

2.3 A ∩B={b,m}

Where b and m are in the middle of both circles.

Translate the following argument into symbolic form. Then determine whether the argument is valid or invalid. If you leave then you lock the door. If you lock the door then you do not go to a friend's house. ​Therefore, if you leave then you go to a friend's house. Write the statement in symbolic form

Write the statement in symbolic form. p right arrow p → q q → ~r ... p → r Is the given argument valid or​ invalid? Invalid

Translate the following argument into symbolic form. Then determine whether the argument is valid or invalid. If the train is on time then I will get to work before you. If I will get to work before you then you will not get in trouble. ​Therefore, if the train is on time then you will get in trouble.

Write the statement in symbolic form. p → q q → ~r ... p → r Is the given argument valid or​ invalid? Invalid

3.5 Translate the following argument into symbolic form. Then determine whether the argument is valid or invalid. If the train is on time, then I will get to work by 9:00. I do not get to work by 9:00. ​Therefore, the train is not on time.

Write the statement in symbolic form. p → q ~q ...~p Is the given argument valid or​ invalid? Valid

A bag has 4 ​red, 3 ​green, and 6 orange balls. If a ball is randomly selected from the​ bag, what is the theoretical probability that it is the color specified in parts​ (a) through​ (d). ​a) green ​b) not green ​c) orange ​d) not orange

a) 3/13 b) 10 / 13 c) 6 / 13 d) 7 / 13

The Greens are moving. Their real estate agent located 77 houses listed for sale in their price range. Of those houses listed for​ sale, 46 had a finished basement. 49 had a​ three-car garage. 34 had a finished basement and a​ three-car garage.

a) How many had a finished basement but not a​ three-car garage? 121 ​b) How many had a​ three-car garage but not a finished​ basement? 15 ​c) How many had either a finished basement or a​ three-car garage? 61

2.5 A survey of 117 high school students determined whether they used​ Instagram, Twitter, or Facebook. The provided information was determined. 72 used Instagram. 58 used Twitter. 77 used Facebook. 34 used Instagram and Twitter. 49 used Instagram and Facebook. 40 used Twitter and Facebook. 24 had all three features.

a) How many of the students surveyed used only​ Instagram? 49-24=25 34-24=10 72-s5-10-24=13 ​b) How many of the students surveyed used Instagram and​ Twitter, but not​ Facebook? 34-24=10 c) How many of the students surveyed used Instagram or​ Twitter? 96 d) How many of the students students used Instagram or​ Twitter, but not​ Facebook? 31 e) 51

A magazine surveyed​ subscribers, asking which of the following reality shows they watched on a regular​ basis: Survivor, The​ Voice, America's Got Talent. The results showed that 356 watched​ Survivor, 294 watched The​ Voice, 284 watched​ America's Got​ Talent, 192 watched Survivor and The​ Voice, 199 watched Survivor and​ America's Got​ Talent, 137 watched The Voice and​ America's Got​ Talent, 66 watched​ Survivor, The​ Voice, and​ America's Got​ Talent, and 26 watched none of these shows. Complete parts ​a) through ​e) below.

a) How many subscribers were​ surveyed? 498 ​b) Of the subscribers​ surveyed, how many watched Survivor and​ America's Got​ Talent, but not The​ Voice? 133 ​c) Of the subscribers​ surveyed, how many watched The​ Voice, but neither Survivor nor​ America's Got​ Talent? 31 ​d) Of the subscribers​ surveyed, how many watched exactly two of these​ shows? 330 ​e) Of the subscribers​ surveyed, how many watched at least one of these​ shows? 472

A magazine surveyed​ subscribers, asking which of the following reality shows they watched on a regular​ basis: Survivor, The​ Voice, America's Got Talent. The results showed that 349 watched​ Survivor, 293 watched The​ Voice, 281 watched​ America's Got​ Talent, 190 watched Survivor and The​ Voice, 196 watched Survivor and​ America's Got​ Talent, 138 watched The Voice and​ America's Got​ Talent, 66watched​ Survivor, The​ Voice, and​ America's Got​ Talent, and 28watched none of these shows. Complete parts ​a) through ​e) below.

a) How many subscribers were​ surveyed? 439 ​b) Of the subscribers​ surveyed, how many watched Survivor and​ America's Got​ Talent, but not The​ Voice? 130 ​c) Of the subscribers​ surveyed, how many watched The​ Voice, but neither Survivor nor​ America's Got​ Talent? 31 d) Of the subscribers​ surveyed, how many watched exactly two of these​ shows? 326 e) Of the subscribers​ surveyed, how many watched at least one of these​ shows? 465

Determine the truth value of the statement (~r ∨ ~p) ∧ q using the following conditions. ​a) p is true​, q is false​, and r is false. ​b) p is true​, q is true​, and r is true.

a) If p is true​, q is false​, and r is false​, what is the value of (~r ∨ ~p) ∧ q​? False ​b) If p is true​, q is true​, and r is true​, what is the value of (~r ∨ ~p) ∧ q​? False

Determine the truth value of the statement (p ∧ ~q) ∧ r using the following conditions. ​a) p is true​, q is true​, and r is true. ​b) p is false​, q is false​, and r is false.

a) If p is true​, q is true​, and r is true​, what is the truth value of (p ∧ ~q) ∧ r​? False b) If p is false​, q is false​, and r is false​, what is the truth value of (p ∧ ~q) ∧ r​? False

A passcode on a smartphone consists of 4 ​digits, and repetition of digits is allowed. ​a) Determine the number of possible four​-digit passcodes. ​b) If a person finds a smartphone and randomly enters 4 ​digits, what is the probability that the correct passcode is​ entered?

a) The number of possible four​-digit passcodes is 10,000 b) The probability that the correct passcode is entered is 1 / 10,000

A set contains thirteen elements. ​a) How many subsets does it​ have? ​b) How many proper subsets does it​ have?

a) The set has 81928192 subsets. ​ ​b) The set has 81918191 proper subsets.

A bag contains six batteries, all of which are the same size and are equally likely to be selected. Each battery is a different brand. If you select three batteries at​ random, use the counting principle to determine how many points will be in the sample space if the batteries are selected

a) With replacement The size of the sample space is 216 sample points.(6*6*6) ​b) Without replacement The size of the sample space is 120 nothing sample points. (6*[6-1}*[6-2])

A disc jockey​ (DJ) has 8 songs to play. Five are slow​ songs, and 3 are fast songs. Each song is to be played only once. ​a) In how many ways can the DJ play the 8 songs if the songs can be played in any​ order? ​b) In how many ways can the DJ play the 8 songs if the first song must be a slow song and the last song must be a slow​ song? ​c) In how many ways can the DJ play the 8 songs if the first two songs must be fast​ songs?

a)40,320 b)14,400 c) 4320

At a music​ festival, there are five bands scheduled to​ play, numbered 1 through 5. a. How many different ways can these bands be arranged to​ perform? b. If band 2 is performing first and band 3 ​last, then how many ways can their appearances be​ scheduled?

a. There are 120 different ways to arrange the bands. If band 2 is performing first and band 3 ​last, there are 6 different ways to arrange the bands.

8.1 An angle whose measure is less than 90° is called​ a(n)

acute angle.

2.1 The two ways to indicate an empty set are_____ and______.

by { } or ∅.

8.3 In a right​ triangle, the side that is opposite the right angle is called the

hypotenuse.

Determine whether the following set is finite or infinite. A=The set of multiples of 4 greater than 24 Is the set A finite or​ infinite?

infinite

Determine whether the set is finite or infinite. The set of odd numbers greater than 11 Which of the following is the correct set​ description?

infinite

Determine whether the argument to the right is valid or invalid. You may compare the argument to a standard form or use a truth table. p ↔ q q → r ... p ∨ r Is the argument valid or​ invalid?

invalid

Two coins are tossed. Assume that each event is equally likely to occur. ​a) Use the counting principle to determine the number of sample points in the sample space. ​b) Construct a tree diagram and list the sample space. ​c) Determine the probability that no tails are tossed. ​d) Determine the probability that exactly one tail is tossed. ​e) Determine the probability that two tails are tossed. ​f) Determine the probability that at least one tail is tossed.

na) There​ is/are 4 sample​ point(s) in the sample space. B. ​HH, HT,​ TH, TT ​c) The probability that no tails are tossed is one fourth d) The probability that exactly one tail is tossed is 1/2 ​e) The probability that two tails are tossed is 1/4 ​f) The probability that at least one tail is tossed is 3/4

3.2 The negation​ ~p will always have the opposite truth value of p.

opposite

3.2 Translate the following statement into symbols. Then construct a truth table for the compound statement and indicate under what conditions the compound statement is true. You did the laundry and you did not leave the garage a mess. ​p: You did the laundry. ​q: You left the garage a mess.

p q =p ∧ ~q T T =F T F =T F T =F F F =F The statement is true whenever p is true and q is false​, and is false otherwise.

For the argument​ below, perform the following. ​a) Translate the argument into symbolic form. ​b) Use a truth table to determine whether the argument is valid or invalid. ​(Ignore differences in​ past, present, and future​ tense.) If she calls me, I'll tell her the news. She calls me. therefore

p → q p therefore q The argument is valid because the conclusion is a tautology.

If we study longer, then we will do well on the test. We will do well on the test. therefore We studied longer. ​a) Let p be ​"We study longer​" and let q be ​"We will do well on the test.​" Write the argument in symbolic form. Choose the correct answer below.

p → q q therefore p The argument is invalid because the argument is an example of the Fallacy of the Converse.

3.5 For the argument​ below, perform the following. ​a) Translate the argument into symbolic form. ​b) Determine if the argument is valid or invalid. Compare the argument to a standard form or use a truth table. If you work overtime, then we will finish the project. We will finish the project. therefore You will work overtime.

p → q q therefore... p The argument is invalid because the argument is an example of the Fallacy of the Converse.

Translate the following argument into symbolic form. Then determine whether the argument is valid or invalid. If the band plays rock music then the band has a lead singer. If the band has a lead singer then the band is not well liked. ​Therefore, if the band plays rock music then the band is well liked. Write the statement in symbolic form.

p → q q → ~r ... p → r Invalid

Translate the following argument into symbolic form. Then determine whether the argument is valid or invalid. If it is snowing then the air is cold outside. If the air is cold outside then it is not winter. ​Therefore, if it is snowing then it is winter.

p → q q → ~r ... p → r Is the given argument valid or​ invalid? Invalid

3.5 For the argument​ below, perform the following. ​a) Translate the argument into symbolic form. ​b) Use a truth table to determine whether the argument is valid or invalid. ​(Ignore differences in​ past, present, and future​ tense.) If the church bell rings, it's noon. It's not noon. therefore

p → q ​~q therefore... ​~p The argument is valid because the statement is a tautology

3.5 For the argument​ below, perform the following. ​a) Translate the argument into symbolic form. ​b) Use a truth table to determine whether the argument is valid or invalid.​ (Ignore differences in​ past, present, and future​ tense.) I tell you something and it's true. If it's true, then you should do what I say. therefore.. if you should do what i say, then i told you something

p ∧ q q → r therefore... r → p The argument is valid because the truth table indicates the conditional statement to be a tautology.

For the argument​ below, perform the following. ​a) Translate the argument into symbolic form. ​b) Determine if the argument is valid or invalid. Compare the argument to a standard form or use a truth table. I will go to college or I will go to trade school. I will not go to college. therefore I will go to trade school.

p ∨ q ~p therefore q ​b) Is the argument valid or​ invalid? The argument is valid because it is an example of Disjunctive Syllogism.

For the argument​ below, perform the following. ​a) Translate the argument into symbolic form. ​b) Determine if the argument is valid or invalid. Compare the argument to a standard form or use a truth table. You will do what I say, or you will be punished. You will not do what I say. therefore You will be punished. ​a) Let p be ​"You will do what I say​" and let q be ​"You will be punished.​" Write the argument in symbolic form. Choose the correct answer below.

p ∨ q ~p therefore q ​b) Is the argument valid or​ invalid? The argument is valid because it is an example of Disjunctive Syllogism.

8.2 A closed figure in a plane determined by three or more straight line segments is called a​ _______.

polygon.

8.2 Two polygons are similar if their corresponding angles have the same measure and the lengths of their corresponding sides are in

proportion.

3.2 Construct a truth table for the given statement. q∧~q Fill in the truth table.

q ~q q∧~q T F F F T F

3.3 Is the statement (p∧~q)∧(~p∨q) a​ tautology, a​ self-contradiction, or​ neither?

self-contradiction

2.3 Disjoint Sets.

sets that do not have any elements in common

8.1 An angle whose measure is 180° is called​ a(n) _______ angle.

straight

Are A ∪ B' and (A' ∩ B)' equal​ statements?

yes

Express the set in roster form. The set of natural numbers between 14 and 176

{15, 16, 17, 18, ..., 175}

Express the set in roster form. The set of natural numbers between 14 and 190

{15, 16, 17, 18, ..., 189}

List all the subsets of the given set. {nectarine, apple, kiwifruit} Choose the answer that lists all of the subsets of {nectarine, apple, kiwifruit}.

{}, ​{nectarine​}, ​{apple​}, ​{kiwifruit​}, ​{nectarine​, apple​}, ​{nectarine​, kiwifruit​}, ​{apple​, kiwifruit​}, ​{nectarine​, apple​, kiwifruit​}

3.1 Let p and q represent the following simple statements. ​p: This is a turtle. ​q: This is a reptile. Write the following compound statement in symbolic form. If this is not a turtle​, then this is not a reptile. The compound statement written in symbolic form is

~p→~q.

3.4 Given the conditional statement p →q, the contrapositive of the conditional statement in symbolic form is​ __________.

~q → ~p

3.1 Let p and q represent the following statements. ​p:Cauliflower is a vegetable. ​q:A triangle has four sides. Express the following statement symbolically. Cauliflower is not a vegetable. ​Symbolically, the statement is______

​Symbolically, the statement is ~p.

A box contains three cards. On one card there is a triangle ​(T​), on another card there is a moon ​(M​), and on the third card there is a rectangle ​(R​). Two cards are to be selected at random with replacement. Complete parts​ (a) through​ (e) below.

​a) Determine the number of sample points in the sample space. There are 9 points in the sample space. c) TT​, TM​, TR​, MT​, MM​, MR​, RT​, RM​, RR Determine the probability that two triangles are selected. The probability is 1/9

In a survey of employees at a fast food​ restaurant, it was determined that 12 cooked​ food, 17 washed​ dishes, 23 operated the cash​ register, 7 cooked food and washed​ dishes, 4 cooked food and operated the cash​ register, 7 washed dishes and operated the cash​ register, 2 did all three​ jobs, and 7 did none of these jobs. Complete parts ​a) through f​) below.

​a) How many employees were​ surveyed? 43 ​(Simplify your​ answer.) ​b) How many of the employees only cooked​ food? 3 ​(Simplify your​ answer.) ​c) How many of the employees only operated the cash​ register? 14 ​(Simplify your​ answer.) ​d) How many of the employees washed dishes and operated the cash register but did not cook​ food? 5 ​(Simplify your​ answer.) ​e) How many of the employees washed dishes or operated the cash register but did not cook​ food? 24 ​(Simplify your​ answer.) ​f) How many of the employees did at least two of these​ jobs? 14 ​(Simplify your​ answer.)

The Greens are moving. Their real estate agent located 81 houses listed for sale in their price range. Of those houses listed for​ sale, 44 had a finished basement. 51 had a​ three-car garage. 33 had a finished basement and a​ three-car garage.

​a) How many had a finished basement but not a​ three-car garage? 11 ​b) How many had a​ three-car garage but not a finished​ basement? 18 ​c) How many had either a finished basement or a​ three-car garage? 62

The Greens are moving. Their real estate agent located 76 houses listed for sale in their price range. Of those houses listed for​ sale, 44 had a finished basement. 49 had a​ three-car garage. 33 had a finished basement and a​ three-car garage. Complete parts ​a) through ​c).

​a) How many had a finished basement but not a​ three-car garage? 11 ​b) How many had a​ three-car garage but not a finished​ basement? 16 ​c) How many had either a finished basement or a​ three-car garage? 60

A magazine surveyed​ subscribers, asking which of the following reality shows they watched on a regular​ basis: Survivor, The​ Voice, America's Got Talent. The results showed that 359 watched​ Survivor, 294 watched The​ Voice, 286 watched​ America's Got​ Talent, 194 watched Survivor and The​ Voice, 201 watched Survivor and​ America's Got​ Talent, 140 watched The Voice and​ America's Got​ Talent, 68 watched​ Survivor, The​ Voice, and​ America's Got​ Talent, and 27 watched none of these shows. Complete parts ​a) through ​e) below.

​a) How many subscribers were​ surveyed? 499 ​b) Of the subscribers​ surveyed, how many watched Survivor and​ America's Got​ Talent, but not The​ Voice? 133 ​c) Of the subscribers​ surveyed, how many watched The​ Voice, but neither Survivor nor​ America's Got​ Talent? 28 ​d) Of the subscribers​ surveyed, how many watched exactly two of these​ shows? 331 ​e) Of the subscribers​ surveyed, how many watched at least one of these​ shows? 472

Determine the truth value of the statement (~p ∨ q) ∧ ~r using the following conditions. ​a) p is false​, q is true​, and r is false. ​b) p is true​, q is false​, and r is true.

​a) If p is false​, q is true​, and r is false​, what is the value of (~p ∨ q) ∧ ~r​? True ​b) If p is true​, q is false​, and r is true​, what is the value of (~p ∨ q) ∧ ~r​? False

3.2 Determine the truth value of the statement (~p ∨ q) ∧ [~(p ∧ ~r)] using the following conditions. ​a) p is true​, q is true​, and r is false. ​b) p is true​, q is false​, and r is true.

​a) If p is true​, q is true​, and r is false​, what is the value of (~p ∨ q) ∧ [~(p ∧ ~r)]​? False ​b) If p is true​, q is false​, and r is true​, what is the value of (~p ∨ q) ∧ [~(p ∧ ~r)]​? False

Determine the truth value of the statement (~r ∧ ~p) ∨ q using the following conditions. ​a) p is true​, q is true​, and r is false. ​b) p is false​, q is false​, and r is true.

​a) If p is true​, q is true​, and r is false​, what is the value of (~r ∧ ~p) ∨ q​? True ​b) If p is false​, q is false​, and r is true​, what is the value of (~r ∧ ~p) ∨ q​? false

Use roster form to write the elements of the following​ set, or state that the set has no elements. The days of the week that begin with S Choose the correct answer below.

​{Saturday, Sunday​}

2.4 Complete​ DeMorgan's laws. ​a) (A ∪ B)′=​____ ​b) (A ∩ B)′=​____

​a) (A ∪ B′= A′ ∩ B′ b) (A ∩ B)′= A' ∪ B′

2.2 Example of a proper subset A c B

A = {jazz,pop,hip hop} B= {classical,jazz,pop,rap,hip hop}

2.1 A Set

A collection of objects is called a set.

3.1 Decide whether the following statement is compound. If it is a compound​ statement, indicate whether it is a​ negation, conjunction,​ disjunction, conditional, or biconditional by using both the word and its appropriate symbol. Mark's friend loves soup for dinner. A)Is the statement a compound​ statement? B)If the statement is​ compound, is it a​ negation, conjunction,​ disjunction, conditional, or​ biconditional?

A)NO B)The Statement is Simple

3.1 ​Statement: It is cold outside and it is snowing. A)Is the above statement simple or​ compound? B) If the statement is​ compound, is it a​ negation, conjunction,​ disjunction, conditional, or​ biconditional?

A)The statement is a compound statement because it combines two or more simple statements. B) The statement is a conjunction. It is represented by the symbol logical and ∧.

​2.4 a) When constructing a Venn diagram with three overlapping​ sets, region​ _____ is generally completed first. ​b) When constructing a Venn diagram with three overlapping​ sets, after completing the region answered in part​ a), the next regions generally completed are​ II, IV, and​ _____.

A)V B)VI

2.1 Choose the correct description of the set. A=​{3​, 6​, 9​, 12​,...} Choose the correct answer below.

A=​{x | x ∈ N and x is a multiple of 3​}

Thirty-seven cities were researched to determine whether they had a professional sports​ team, a​ symphony, or a​ children's museum. Of these​ cities, 19 had a professional sports​ team, 23 had a​ symphony, 5 had a​ children's museum, 13 had a professional sports team and a​ symphony, 7 had a professional sports team and a​ children's museum, 11 had a symphony and a​ children's museum, and 5 had all three activities.

I=4 II= 8 III=4 IV= 2 V= 5 VI=6 VII=2 VIII=6 Let A represent the cities that had a professional sports team. Let B represent the cities that had a symphony. Let C represent the cities that had a​ children's museum. Note that 5 had all three activities. This group is represented on the Venn diagram by region V. Next, note that 11 had a symphony and a​ children's museum. This group is represented on the Venn diagram by regions V and VI. Since region V is known to contain 5​, region VI must contain 11-5, or 6. The problem states that 7 had a professional sports team and a​ children's museum. This group is represented on the Venn diagram by regions IV and V. Since region V is known to contain 5​, region IV must contain 7−5​, or 2

2.4 A Venn diagram contains three​ sets, A,​ B, and​ C, as shown. If region V contains 9 elements and there are 17 elements in A∩C​, how many elements belong in region IV​? Explain.

There​ is/are 8 ​element(s) in region IV because the number of elements in A∩C is the sum of the number of elements in region V and the number of elements in region IV.

2.3 U is the set of colleges in a country. A is the set of colleges that have a basketball team. B is the set of colleges that have a physics department. Describe A ∪ B′ in words.

is the set of colleges in a country that have a basketball team or do not have a physics department.

2.2 cont.... List the number of distinct subsets for the set {S,L,E,D}.

{S,L,E,D} {S,L,E} {S,L,D} {S,E,D} {L,E,D} {S,L,} {S,E,} {S,D} {L,E} {L,D} {E,D} {S} {L} {E} {D} {} or 0 with / in it.

3.1 Select the correct choices that complete the sentences below. ​a) The negation is symbolized by​ ~ and is read​ "_______." ​b) The conjunction is symbolized by ∧ and is read​ "_______." ​c) The disjunction is symbolized by ∨ and is read​ "_______."

A) Not B) And C) Or

3.1 Statement: A polygon is a quadrilateral if and only if it has four sides. A) Is the above statement simple or​ compound? B)If the statement is​ compound, is it a​ negation, conjunction,​ disjunction, conditional, or​ biconditional?

A) The statement is a compound statement because it combines two or more simple statements. B)The statement is a biconditional. It is represented by the symbol left right arrow↔.

A universal set U consists of 23 elements. If sets​ A, B, and C are proper subsets of U and ​ n(U)=23​, ​n(A ∩ ​B)=​n(A ∩ ​C)=​n(B ∩ ​C)=8​, ​ n(A ∩ B ∩ ​C)=1​, n(A ∪ B ∪ ​C)=22​, determine each of the following. ​a)​ n(A ∪ ​B) ​b) n(A′ ∪ C) ​c) n(A ∩ B)′

Construct a Venn diagram with three overlapping​ circles, where one circle represents set​ A, one circle represents set​ B, and one circle represents set C. Fill in the regions using the given information by working from the center of the diagram outward. Begin with region​ V, the intersection of all three​ sets denoted A∩B∩C. According to the problem​ statement, there is 1 element in region V. Determine the number of elements in region II. Note that regions II and V combine to form the intersection of sets A and​ B, denoted A∩B. This means that subtracting​ n(A∩B∩C) from​ n(A∩B) will result in the number of elements in region II. Find the number of elements in region II. ​n(A ∩ ​B)−​n(A ∩ B ∩ ​C)=8−1 ​n(A ∩ ​B)−​n(A ∩ B ∩ ​C)= 7 Determine the number of elements in region IV. Note that regions IV and V combine to form the intersection of sets A and​ C, denoted A ∩ C. This means that subtracting​ n(A ∩ B ∩ ​C) from​ n(A∩​C) will result in the number of elements in region IV. Find the number of elements in region IV. ​n(A ∩ ​C)−​n(A ∩ B ∩ ​C)=8−1 ​n(A ∩ ​C)−​n(A ∩ B ∩ ​C)= 7 Determine the number of elements in region VI. Note that regions V and VI combine to form the intersection of sets B and​ C, denoted B ∩ C. This means that subtracting​ n(A ∩ B ∩ ​C) from​ n(B∩​C) will result in the number of elements in region VI. Find the number of elements in region VI. ​n(B ∩ ​C)−​n(A ∩ B ∩ ​C)=8−1 ​n(B ∩ ​B)−​n(A ∩ B ∩ ​C)= 7 Now determine the number of elements in regions​ I, III, and VII. Note that the total number of elements in regions​ I, III, and VII combined is equal to the total number of elements in regions​ II, IV,​ V, and VI combined subtracted from the total number of elements in sets​ A, B, and C​ combined, a set denoted A ∪ B ∪ C. Find the number of elements in regions​ II, IV,​ V, and VI combined. 7+7+1+7= 22 Since the total number of elements in regions​ II, IV,​ V, and VI is 22​, and it is given that the total number of elements in sets​ A, B, and​ C, n(AUB∪C), is 22​, this leaves 22−22=0 elements left in regions​ I, III, and VII combined. If there are no elements in all three regions​ combined, then there are no elements in each individual region.​ Thus, there are 0 elements in each of region​ I, region​ III, and region VII. ​Finally, determine the number of elements in region VIII. The number of elements in region VIII is equal to the number of elements in the union of sets​ A, B, and C subtracted from the number of elements in U. Find the number of elements in region VIII. n(U)−​n(A∪B∪C)=23−22=1 a) To determine​ n(A∪B), count the number of elements that are in set A or are in set B. This number can be found by finding the sum of the numbers of elements in regions​ I, II,​ III, IV,​ V, and VI. Find the number of elements in set A or in set B. ​n(A ∪ ​B)=0+7+0+7+1+7 ​n(A ∪ ​B)= 12 ​b) To determine nA′ ∪ C​, count the number of elements that are not in set A or are in set C. This number can be found by finding the sum of the numbers of elements in regions​ III, IV,​ V, VI,​ VII, and VIII. Find the number of elements not in set A or in set C. nA′ ∪ C=0+7+1+7+0+1 nA′ ∪ C= 16 c) To determine n(A ∩ B)′ ​, count the number of elements that are not in the intersection of sets A and B. The intersection of sets A and B is comprised of regions II and​ V, so this number can be found by finding the sum of the numbers of elements in regions​ I, III,​ IV, VI,​ VII, and VIII. Find the number of elements not in the intersection of sets A and B. n(A ∩ B)′=0+0+7+7+0+1 n(A ∩ B)′=15

2.5 At a community​ college, a survey was taken to determine where students study on campus. Of the 250 students​ surveyed, it was determined that 168 studied in the library. 135 studied in the cafeteria. 80 studied in both the library and the cafeteria. ​a) Of those students​ surveyed, how many studied only in the​ library?

Let A represent the number of students who studied in the library. Let B represent the number of students who studied in the cafeteria. Note that 78 students studied in both the library and the cafeteria. Region II on the Venn diagram represents this group. Therefore, 78 students studied in both the library and the cafeteria. ​Next, note that 176 students studied in the library. Regions I and II on the Venn diagram represent this group. Since region II is known to contain 78​,region I must contain a total of 176−78​,or 98. Therefore, 98 students studied only in the library. ​b) Of those students​ surveyed, how many studied only in the​ cafeteria? Note that 143 students studied in the cafeteria. Regions II and III on the Venn diagram represent this group. Since region II is known to contain 78​, region III must contain a total of 143−78​,or 65. Therefore, 65 students studied only studied in the cafeteria. The number of students who only studied in the cafeteria is shown on the diagram to the right. ​c) Of those students surveyed, how many did not study in either​ place? The problem states that 280 students were surveyed. The number in each of the regions I through III is known. Therefore, region IV must contain 280−98−78−65​, or 39. The completed Venn diagram is shown to the right.​ Therefore, the number of students who studied in neither place is 39.

2.3 U is the set of furniture stores . A is the set of furniture stores that sell mattresses. B is the set of furniture stores that sell kitchen tables. C is the set of furniture stores that sell outdoor furniture. Describe A ∩ B ∩ C in words.

The set of furniture stores that sell mattresses​, kitchen tables​, and outdoor

2.3 Let U represent the set of students in a math class. Let A represent the set of students in the class that have an A in the class. Describe A′.

The set of students in the math class that do not have

2.3 Complement

The set of all elements in the universal set that are not in set A is called the complement of set A.

2.3 Let U be the set of furniture​ stores, A be the set of all furniture stores that sell outdoor furniture​, B be the set of all furniture stores that sell tables​, and C be the set of all furniture stores that sell metal furniture. Describe the set B ∪ C in words.

The set of all furniture stores that sell tables or metal furniture.

2.3 Cartesian

The set of all possible ordered pairs of the form​ (a,b), where a∈A and b∈​B, is called the Cartesian product of set A and set B.

2.3 Difference

The set of elements that belong to set​ A, but not to set​ B, is called the difference of two sets A and B.

2.3 Set U is the set of farms in a certain country. Set A is the set of farms in the country that produce peas. Set B is the set of farms in the country that produce beans. Describe the set A ∪ B in words.

The set of farms in the country that produce peas or produce

2.3 U is the set of furniture stores. A is the set of furniture stores that sell TV stands. B is the set of furniture stores that sell bed frames. C is the set of furniture stores that sell cabinets. Describe the set A ∪ B ∪ C in words.

The set of furniture stores that sell TV stands or bed frames or cabinets.

A survey of 112 high school students determined whether they used​ Instagram, Twitter, or Facebook. The provided information was determined. 70 used Instagram. 62 used Twitter. 69 used Facebook. 35 used Instagram and Twitter. 42 used Instagram and Facebook. 38 used Twitter and Facebook.

a) How many of the students surveyed used only​ Instagram? 35-21=14 42-21=21 70-14-21-21=14 ​b) How many of the students surveyed used Instagram and​ Twitter, but not​ Facebook? 35-21=14 ​c) How many of the students surveyed used Instagram or​ Twitter? 38-21=17 62-21-17-14=10 69-21-21-17=10(not in addition portion) 14+14+21+21+17+10=97(answer to c) d) How many of the students students used Instagram or​ Twitter, but not​ Facebook? 14+14+10=38 ​e) How many of the students surveyed used exactly two of the social​ medias? 14+21+17=52

A survey of 123 high school students determined whether they used​ Instagram, Twitter, or Facebook. The provided information was determined. 82 used Instagram. 69 used Twitter. 79 used Facebook. 42 used Instagram and Twitter. 51 used Instagram and Facebook. 45 used Twitter and Facebook. 27 had all three features.

​a) How many of the students surveyed used only​ Instagram? To find the number of students that used only Instagram, work from the center of the diagram outward. Note that 27 students used all three. ​Next, note that 42 students used both Instagram and Twitter. Since region V is known to contain 27​, region II must contain 42−27​,or 15. The problem states that 51 students used Instagram and Facebook. Since region V is known to contain 27​,region IV must contain 51−27​,or 24. The problem states 82 students used Instagram. Since region II is known to contain 15​,region IV is known to contain 24​, and region V is known to contain 27​, region I must contain 82−15−24−27​,or 16. ​So, 16 students surveyed used only Instagram. ​b) How many of the students surveyed used Instagram and Twitter but not​ Facebook? The number of student who used Instagram and Twitter but not Facebook is represented by region II in the diagram. So, 15 students surveyed used Instagram and Twitter but not Facebook ​c) How many of the students surveyed used Instagram or​ Twitter? Note that 45 students used Twitter and Facebook. Regions V and VI on the Venn diagram represent this group. Since region V is known to contain 27​, region VI must contain 45−27​, or 18. The problem states 69 students used Twitter. Since region II is known to contain 15​,region V is known to contain 27​,and region VI is known to contain 18​,region III must contain 69−15−27−18​,or 9. The problem states that 79 students used Facebook. Since region IV is known to contain 24​, region V is known to contain 27​, and region VI is known to contain 18​, region VII must contain 79−24−27−18​, or 10. The number of students who used Instagram or Twitter is represented by all the regions in circle A and circle B. Add the number of resorts that are in region I through region VI. 16+15+9+24+27+18=109. ​So, 109 of the students surveyed used Instagram or Twitter. ​d) How many of the students surveyed used Instagram or Twitter but not​ Facebook? To find the number of students surveyed used Instagram or Twitter but not​ Facebook, find the number of students in regions​ I, II, and III on the Venn diagram. Add the number of resorts that are in region I through region III. 16+15+9=40 So, 40 of the students surveyed used Instagram or Twitter but not Facebook. e) How many of the students surveyed used exactly two of the social​ medias? Notice that regions​ II, IV, and VI on the Venn diagram represent students who used exactly two of the social medias. Add the number of students that are in regions​ II, IV, and VI. 15+24+18=57 ​So, 57 students surveyed used exactly two of the social medias.

2.5 At a community​ college, a survey was taken to determine where students study on campus. Of the 250 students​ surveyed, it was determined that 170 studied in the library. 127 studied in the cafeteria. 77 studied in both the library and the cafeteria.

​a) Of those students​ surveyed, how many studied only in the​ library? 170-77=93 ​b) Of those students​ surveyed, how many studied only in the​ cafeteria? 127-77=50 Of those students​ surveyed, how many did not study in either of these​ places? 250-93-50-77=30

Use the set B=​{3, 5, 7, 9, 12​} to determine ​n(B​).

​n(B​) ​=5

2.2 List all the subsets of the given set. ​{sheep​, horse​}

​{ }, ​{sheep​}, ​{horse​}, ​{sheep​, horse​}

2.1 Use roster form to write the elements of the following​ set, or state that the set has no elements. The days of the week that begin with T

​{Tuesday, Thursday​}