math chapter 7
List the positive integers that are divisors of 12. The positive integers that are divisors of 12 are 1,2,3,4,6,12.
7.1 question 1, pic 1
The following proposition is given. If n is an integer that is a multiple of 21, then n is an integer that is a multiple of 3 and a multiple of 7.
7.1 question 10, pics 14-15
Construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. p--> not q
7.1 question 11, pics 16-19
Construct a truth table for the proposition and determine whether the proposition is a contingency, tautology, or contradiction. (q-->p) ^ q
7.1 question 12, pics 20-22
Construct a truth table for the following proposition. ¬p→(p∧q)
7.1 question 13, pics 23-26
Construct a truth table for the following proposition. (¬p∧q) ∨ (q→p)
7.1 question 14, pics 27-30
Construct a truth table to verify the implication. not p--->(q ^ not q)
7.1 question 15, pics 31-34
Complete the truth table to verify the equivalence. ¬p → (p ∨ q) is equivalent to (p ∨ q) true
7.1 question 16, pic 35
Construct a truth table to verify the following equivalence. p∧(p→q)≡q∧(q→p)
7.1 question 17, pics 36-38
Verify the equivalence using formulas. ¬(p→¬q) ≡ p ∧ q equal
7.1 question 18, pics 39-40
List the positive multiples of 17 that are less than 100. The positive multiples of 17 that are less than 100 are 17,34,51,68,85.
7.1 question 2, pic 2
Express the proposition, ¬q, in an English sentence, and determine whether it is true or false, where p and q are the following propositions. p: "75 is prime" q: "75 is odd" answer: "75 is not odd" and "false"
7.1 question 3, pics 3-4
Express the proposition, the converse of p→q, in an English sentence, and determine whether it is true or false, where p and q are the following propositions. p: "51 is prime" q: "51 is odd" answer: "If 51 is odd, then 51 is prime." and "false"
7.1 question 4, pics 5-6
Express the proposition r∨s in an English sentence, and determine whether it is true or false, where r and s are the following propositions. r: "29+28+27 is greater than 896" s: "8•102+9•10+6 equals 896"
7.1 question 5, pics 7-8
Describe the proposition as a negation, disjunction, conjunction, or conditional, and determine whether the proposition is true or false. If −11<0, then (−11)^2<0.
7.1 question 7, pics 9-11
Describe the proposition as a negation, disjunction, conjunction, or conditional, and determine whether the proposition is true or false. 18 is not prime.
7.1 question 8, pic 12
Describe the proposition as a negation, disjunction, conjunction, or conditional, and determine whether the proposition is true or false. 6 is odd or 6 is prime.
7.1 question 9, pic 13
Is the set of perfect cubes a subset of the set of integers?
7.2 question 1, pics 1-3
For P = {3, 12, 14, 15}, Q = {1, 7, 13}, and R = {3, 7, 8, 13}, find P ∪ (Q ∩ R).
7.2 question 10, pic 12
Consider the set N of positive integers to be the universal set, and let E=n∈N n is even and P=n∈N n is prime. Determine whether E∪P is finite or infinite
7.2 question 11, pics 13-14
Consider the set N of positive integers to be the universal set, and let O=n∈N n is odd and P=n∈N n is prime. Determine whether O∩P is finite or infinite.
7.2 question 12, pics 15-16
Consider the set N of positive integers to be the universal set. Sets H, T, E, and P are defined to the right. Determine whether or not the sets H ' and E are disjoint.
7.2 question 13, pic 17
Determine if the statement below is true or false. If A⊂B, then A∩B=B.
7.2 question 14, pics 18-19
Find the indicated number of elements by referring to the table of enrollments in a finite mathematics class. Let the universal set U be the set of all 113 students in the class, A the set of students from the College of Arts & Sciences, B the set of students from the College of Business, F the set of freshman, and S the set of sophomores. Find the number of students in A.
7.2 question 15, pic 20
Find the indicated number of elements by referring to the table of enrollments in a finite mathematics class. Let the universal set U be the set of all 127 students in the class, A the set of students from the College of Arts & Sciences, B the set of students from the College of Business, F the set of freshman, and S the set of sophomores. Find the number of students in A∩F.
7.2 question 16, pic 21
The management of a company, a president and three vice-presidents, denoted by the set P, V1, V2, V3, wish to select a committee of 3 people from among themselves. How many ways can this committee be formed? That is, how many 3-person subsets can be formed from a set of 4 people
7.2 question 17, pic 22
When receiving a blood transfusion, a recipient must have all the antigens of the donor. A person may have one or more of the three antigens A, B, and Rh, or none at all. Eight blood types are possible, as indicated in the Venn diagram, where U is the set of all people under consideration. Using the Venn Diagram, indicate which of the eight blood types are included in A∩B.
7.2 question 18, pics 23-25
When receiving a blood transfusion, a recipient must have all the antigens of the donor. A person may have one or more of the three antigens A, B, and Rh, or none at all. Eight blood types are possible, as indicated in the Venn diagram, where U is the set of all people under consideration. Using the Venn Diagram, indicate which of the eight blood types are included in B'∩A.
7.2 question 19, pics 26-27
If the universal set is the set of rational numbers, is the set of negative rational numbers the complement of the set of positive rational numbers?
7.2 question 2, pic 4
Indicate whether the statement is true or false. {a,e,j, k}={j, k, e, a} true
7.2 question 3, pic 5
Write the resulting set using the listing method. {1,6} ∩ {1,7,8} {1,6} ∩ {1,7,8}=1
7.2 question 4, pic 6
Find the union of the sets. {2, 4, 6, 8}∪ {1, 3, 5, 7} ={1, 2, 3, 4, 5, 6, 7, 8}
7.2 question 5, pic 7
Write the resulting set using the listing method. An even number between 2 and 10. {2,4,6,8,10}
7.2 question 6, pic 8
Let U={2, 3, 4, 5, 6, 7, 8} and A={2, 5, 6, 7}. Find A′.
7.2 question 7, pic 9
Refer to the Venn diagram below. How many elements are in the set A'?
7.2 question 8, pic 10
Refer to the Venn diagram to the right. How many elements are in the following set? (AUB)'
7.2 question 9, pic 11
Determine the validity of the following statements. (A) If A or B is the empty set, then A and B are disjoint. -true B) If A and B are disjoint, then A or B is the empty set. -The statement is false. For example, A could be the set of odd numbers and B could be the set of even numbers.
7.3 question 10, pic 15
You would like to make a salad that consists of lettuce, tomato, cucumber, and radishes. You go to the supermarket intending to purchase one variety of each of these ingredients. You discover that there are nine varieties of lettuce, six varieties of tomatoes, three varieties of cucumbers, and three varieties of radishes for sale at the supermarket. How many different salads can you make?
7.3 question 11, pic 16
A small combination lock on a suitcase has 4 wheels, each labeled with the 10 digits 0 to 9. How many 4 digit combinations are possible if no digit is repeated? If digits can be repeated? If successive digits must be different?
7.3 question 12, pics 17-20
Which of the numbers x, y, z, or w must equal 0 if A⊂B?
7.3 question 13, pics 21-23
A class of 50 music students includes 14 who play the piano, 16 who play the guitar, and 7 who play both the piano and the guitar.
7.3 question 14, pic 24
Management selection. A corporation plans to fill 2 different positions for vice-president, V1 and V2, from administrative officers in 2 of its manufacturing plants. Plant A has 9 officers and plant B has 5.
7.3 question 15, pic 25
A survey of 1100 people in a certain city indicates that 830 own microwave ovens, 750 own DVD players, and 600 own microwave ovens and DVD players. (A) How many people in the survey own either a microwave oven or a DVD player? (B) How many own neither a microwave oven nor a DVD player? (C) How many own a microwave oven and do not own a DVD player?
7.3 question 16, pics 26-30
Politics. If 11,846 people voted for a politician in his first election, 12,658 voted for him in his second election, and 5,408 voted for him in the first and second elections, how many people voted for this politician in the first or second election?
7.3 question 17, pic 31
Solve the following problem two ways: (a) using a tree diagram and (b) using the multiplication principle. How many 3-letter code words can be formed from the first 3 letters of the alphabet if no letter can be used more than once?
7.3 question 4, pics 1-2
A college offers 2 introductory courses in history, 1 in science, 2 in mathematics, 1 in philosophy, and 1 in English.
7.3 question 5, pic 3
The 12 colleges of interest to a high school senior include 8 that are expensive (tuition more than $20,000 per year), 8 that are far from home (more than 200 miles away), and 5 that are both expensive and far from home.
7.3 question 6, pics 4-7
Use the given information to determine the number of elements in each of the four disjoint subsets in the following Venn diagram. n(A) = 20, n(B) = 55, n(A ∩ B) = 10, n(U) =
7.3 question 7, pics 8-10
Use the given information to complete the following table. n(A)=45, n(B)=55 n(A∪B)=85, n(U)=110
7.3 question 9, pics 11-14
Is the selection a permutation, a combination, or neither? A coach chooses 10 players for his team from the students who tried out.
7.4 question 10, pics 18-19
In a horse race, how many different finishes among the first 3 places are possible if 19 horses are running? (Exclude ties)
7.4 question 11, pics 20-21
A) How many ways can a 3-person subcommittee be selected from a committee of 7 people? (B) How many ways can a president, vice-president, and secretary be chosen from a committee of 7 people?
7.4 question 12, pic 22
From a standard 52-card deck, how many 9-card hands contain all spades?
7.4 question 13, pics 23-24
Discuss the validity of each statement. If the statement is true, explain why. If not, give a counter example. If n and r are positive integers and 1<r<n, then nCr=nCn−r.
7.4 question 14, pic 25
How many ways can 4 people sit in a row of 7 chairs?
7.4 question 15, pic 26
An electronics store receives a shipment of 30 graphing calculators, 3 that are defective. Four of the calculators are selected to be sent to a local high school. (A) How many selections can be made? (B) How many of these selections will contain no defective calculators?
7.4 question 16, pic 27
A 4-person grievance committee is to be selected out of 2 departments, A and B, with 11 and 23 people, respectively. In how many ways can the following committees be selected? Complete parts (A) through (E) below.
7.4 question 17, pics 28-29
Evaluate the expression. If the answer is not an integer, round to four decimal places. 7! + 8!
7.4 question 3, pic 1
Evaluate the expression. 19!/4! (19-4)!
7.4 question 4, pics 2-3
Evaluate 12C4.
7.4 question 5, pics 4-5
Evaluate the expression. 6p3
7.4 question 6, pics 6-7
Evaluate the expression. 19c5/39c5
7.4 question 7, pics 8-12
Simplify the expression assuming that n is an integer and n≥3.
7.4 question 8, pic 13
Is the selection below a permutation, a combination, or neither? A man chooses the winner and runner up in a horse race
7.4 question 9, pics 14-17