Chapter 1-4 Discrete Math Final Study Guide

A brick wall has 60 bricks in the first row, but each row has 3 fewer bricks than the previous one. How many bricks are in the 12th row? 27 30 21 33 24

27 an = difference*(term looking for - 1) + first number

Find the sum of the following geometric series: ∞ Σ 2(1/3)n-1 n = 1

3

Suppose you have a drawer full of white socks and black socks. What is the minimum number of socks you would need to pull out of the drawer to guarantee a pair of matching socks? 1 2 4 3

3

A sandwich shop offers 3 types of meat, 3 types of cheeses, 3 types of bread, and 4 types of condiments for its sandwiches. If you select one of each of the four ingredients, how many possible sandwiches can you create? 108 36 9 13

3*3*3*4=108

Express this series using sigma notation: 3 + 5 + 9 + 17 + 33.

# above Σ = # of terms # in front of Σ is d/pattern plug in n and solve

In the sequence 9, 14, 19, 24, 29,.... Say we're using a to describe the terms. What is the value of the term below? a3 14 29 24 19 9

19

How many steps are in mathematical induction? 5 3 4 2 1

2

Which of the following circles would be considered a different arrangement? 3 4 1 2

2

Find the sum of the following geometric series: 9 Σ 4(2)n-1 n = 1 2,048 2,044 174,763 511 1,024

2044

Calculate 4! (factorial) 24 6 12 48 9

24

Which is the correct definition of a conjunction? It is when two statements are connected with an 'AND'. The resulting compound statement can be labeled as true when just one of the statements is true. It is when two statements are connected with an 'OR'. The resulting compound statement can be labeled as true when just one of the statements is true. It is when two statements are connected with an 'AND'; only when both statements are true is the resulting compound statement true. It is when two statements are connected with a 'NOT'. The resulting compound statement can be labeled as true when just one of the statements is true. It is when two statements are connected with an 'OR'; only when both statements are true is the resulting compound statement true.

It is when two statements are connected with an 'AND'; only when both statements are true is the resulting compound statement true. Conjunction use AND disjunctions use OR

Which of the following is the 14th term of the sequence below? 11, 22, 44, 88, 176, 45,056 180,224 8,192 16,384 90,112

90,112 an = a1 * r(n-1) an = missing term/value a1 = first value r = multiple (n = 1st value - 1)

Twenty students compete in a school-wide marathon and each student is of comparable running ability. Of the 20 students, 15 were boys and 5 were girls. What is the probability that boys will place 1st, 2nd, and 3rd in the marathon? 91/228 7/228 73/228 13/228

91/228

Find the sum of the arithmetic series 106 +100 + 94 + ... + 28 + 22. 768 960 1536 896 1,792

960

Find the rule for this series: 2 + 6 + 18 + 54 + ... an = n + 4 an = 2(3) n-1 an = 3(2) n-1 an = n^2 + 2

n1(d)n-1

Given item A, which of the following would be the value of item B? Item A: 5 Σ an = 30 n-1 Item B: 5 Σ 3an - 2 n-1

80

What is the sum of all the even integers from 2 to 250?

15,750 n = (last term - first term / common difference) + 1 S = n/2 * (first + last term)

If a2 = 5 and a8 = 35, what is the value of a30? 185 245 150 145 158

an = dn + n0 145

A locker combination contains four numbers between 1 and 20 and none of the numbers can be repeated. What is the probability that the locker combination will consist of all even numbers? 14/969 28/323 28/969 14/323

14/323

If 55 and 89 are two sequential terms of the Fibonacci sequence, what term would come next? 178 123 110 144 165

144

Jimmy is making multi-flavored ice cream cones by scooping in different flavors one at a time. Jimmy has 6 different flavors but can only put 3 flavors in each cone. The order of the flavors is important to him as it affects how he tastes each ice cream. How many different arrangements of cones can Jimmy make? 60 18 120 30

120

Six people are going to sit at a round table. How many different ways can this be done? 720 60 360 120

120

Solve the expression 5P4 (P = permutation ) 5 120 60 20

120

What is the 83rd term of the sequence 91, 87, 83, 79, ... ( = a1, a2, a3, a4, ...)? -247 -243 -237 -233 -229

-237 d(n-1) + n1 or d*n + n 0 D = difference/pattern= difference from current position to desired position n1 = starting position/beginning

Suppose you have a circular arrangement of three items. If the circle is free, in how many ways can the items be arranged? 1 2 4 3

1

Twenty students compete in a school-wide marathon and each student is of comparable running ability. Of the 20 students, 15 were boys and 5 were girls. What is the probability that girls will place 1st, 2nd, and 3rd in the marathon? 5/120 1/114 5/114 1/120

1/114

Annie writes the numbers 1 through 10 on note cards. She flips the cards over so she cannot see the number and selects three cards from the stack. What is the probability that she has selected the cards numbered 1, 2, and 3? 1/120 3/100 3/7 1/720

1/120

Jimmy has the letters for the state of MISSISSIPPI written on cards, one letter per card. He turns the cards over and mixes up the order. If he selects one card at a time without replacing the cards, what is the probability that he will spell the word MISS in order? 48/495 1/495 3/365 1/165

1/165

A man is deciding what to wear to work. He is considering an outfit from among 5 shirts, 3 pairs of pants, and 7 ties. How many possible outfits are there? 30 105 120 15

105

Which choice below represents an arithmetic series? 10 + 30 + 90 + 270 + 810 5, 25, 125, 625, 3, 123 12 + 10 + 8 + 6 + 4 1 + 4 + 9 + 16 + 25 6, 10, 14, 16, 22

12 + 10 + 8 + 6 + 4

A vehicle license plate uses three numbers and three letters on each plate. The numbers are listed first and then the letters. The numbers used range from 0-9 and the letters used can be any letter of the 26 letters of the alphabet. On any given license plate, the letters can be repeated, but the numbers cannot be repeated. How many different plates are possible? 12,654,720 108 11,232,000 17,576,000

12,654,720

A ball is dropped and begins bouncing. On the first bounce, the ball travels 3 feet. Each consecutive bounce is 1/8 the distance of the previous bounce. What is the total distance that the ball travels after it first hits the ground? Round to the nearest hundredth. 1.10 feet 3.43 feet 24.00 feet 3.75 feet 2.63 feet

3.43 feet

Which group of numbers does not appear to be a sequence with a set pattern? 256, 128, 64, 32, 16, ... 10, 30, 90, 270, 810, ... 13, 6, -1, -8, -15, ... 4, 10, 16, 22, 28, ... 4, -13, 1, 5, 16, ...

4, -13, 1, 5, 16, ...

8 different novels are to be placed side-by-side on a shelf. How many ways can the 8 novels be arranged on the shelf? 56 8 40,320 20,160

40,320

Solve 8! (the factorial of 8) 362,880 5,040 20,160 40,320

40,320

Evaluate the following: 15 Σ 3n + 5 n=4

402

Solve the expression 7P2 (P = permutation) 42 2,640 120 5,040

42

A brick wall contains 52 bricks in its bottom row and 49 bricks in the next row up from the bottom row. Each subsequent row contains 3 fewer bricks than the row immediately below it. If the wall contains 16 rows, how many bricks total make up the wall? 720 944 518 472 360

472

Jane is attempting to unlock her locker but has forgotten her locker combination. The lock uses 3 numbers and includes only the numbers 1 to 9. The digits cannot be repeated in the combination. How many possible locker combinations can be formed? 504 locker combinations 27 locker combinations 252 locker combinations 362,880 locker combinations

504 locker combinations

A ball is dropped from an unknown height (h) and it repeatedly bounces on the floor. After each bounce, the ball reaches a height that is 2/3 of the height from which it previously fell. Which of the following represents the distance the ball bounces from the first to the seventh bounce with sigma notation?

6 Σ 2h(2/3) n n=1

A couple wants to plant some shrubs around a circular walkway. They have seven different shrubs. How many different ways can the shrubs be planted? 720 1440 120 5040

720

A horse race includes 10 participants. How many possible finishes are there for the top three positions: first (win), second (place), and third (show)? 1,000 3,628,800 720 5,040

720

Evaluate the following: 6 Σ 2(3) n-1 n=1

728

Find the sum 1 + 8 + 15 + 22 + 29 using the formula for an arithmetic series. 100 70 29 75

75

All of the following statements could be proven with a direct proof EXCEPT: If x and y are integers and x is odd and y is even, then xy is even. If q is an even integer, then q^2 is an even integer. If m is an even integer, then -m is an even integer. If n is an odd integer, then m is an even integer. If c is an even integer, then 5c + 6 is even.

If n is an odd integer, then m is an even integer.

Which of the following statements is FALSE? The element which is not related to any other element in a Hasse diagram is called maximal element. A POSET is called a meet semilattice if every pair of elements has a 'least upper bound' element. A POSET is called a lattice if it is both a join semilattice and meet semilattice. A relation R, over a set A, is reflexive if every element of the set is related to itself.

A POSET is called a meet semilattice if every pair of elements has a 'least upper bound' element. Join = Least upper bound, meet = greater lower bound lattice = both

Chapter 3 Lesson 5: What's the difference between a sequence and a series? A sequence is an ordered list of numbers, and a series is the sum of a sequence's terms. A series is an ordered list of numbers, and a sequence is the sum of a series' terms. A series can be infinite, but a sequence must always be finite. There is no difference because both terms describe the same mathematical expression.

A sequence is an ordered list of numbers, and a series is the sum of a sequence's terms.

The set of all elements that are under consideration for a particular problem or situation is known as: A complement set A notation set A universal set A subset

A universal set/union

Which of the following occurs with a direct proof? Statements are supported by known facts and definitions. A series of statements are made. All are correct A conditional statement is proven.

All are correct

Which of these questions should you ask yourself to help determine if mathematical induction is a good method to prove a given statement? Can I prove the first few cases easily? If the statement is true for the first k elements, can we use that to show it is true for the (k+1)st element? Am I trying to prove something is true for an infinite set of elements? All of these are questions that could be asked when determining if mathematical induction is a good method of proof to use to prove a statement.

All of these are questions that could be asked when determining if mathematical induction is a good method of proof to use to prove a statement.

Which of the following statements about the pigeonhole principle is TRUE? All of these statements about the pigeonhole principle are true. The pigeonhole principle states that if we are placing pigeons in pigeonholes, and there are more pigeons than pigeonholes, it must be the case that at least one pigeonhole has more than one pigeon in it. If we are placing 7 pigeons in 5 pigeonholes, it must be the case that at least one of the pigeonholes has more than one pigeon in it. The pigeonhole principle is a simple yet extremely useful concept used in combinatorics.

All of these statements about the pigeonhole principle are true.

The name one-to-one describes which function? An injective function A surjective function A bijective function An exponential function

An injective function

Which type of logical fallacy is illustrated in the example below? I will get an A in math class because I spent many sleepless nights studying. Appeal to emotion Appeal to ignorance Appeal to popularity Appeal to wisdom

Appeal to emotion

Suppose we wanted to use mathematical induction to prove that for each natural number n, 2 + 5 + 8 + ... + (3n - 1) = n(3n - 1) / 2. In our induction step, what would we assume to be true and what would we show to be true. We would show that if the statement is true for the first k elements, then it is true for the (k + 1)st element. None of these are correct. We would show that 2 - 1 = 1. We would show that the statement was true for n = 1 and for n = 2 by plugging 1 and 2 into our formula separately, and making sure they both make a true statement.

Assume: 2 + 5 + 8 + ... + (3k - 1) = k(3k - 1) / 2 Show: 2 + 5 + 8 + ... + (3k - 1) + (3(k +1) - 1) = (k + 1)(3(k+1) - 1) / 2

Which of the following is NOT necessary for a relation to be called a partially ordered relation? Asymmetric relation Transitive relation Reflexive relation Anti-symmetric relation

Asymmetric relation

If you're placing pigeons in pigeonholes, and there are more pigeons than pigeonholes, which of the following must be TRUE? The pigeons that have the same coloring will need to be placed together. All of the pigeons will need to be placed in one hole. No pigeonhole will contain more than one pigeon. At least one pigeonhole will contain more than one pigeon.

At least one pigeonhole will contain more than one pigeon.

Select the appropriate truth table for the tautology p --> (p v q)

Column must state "Truth Table for p --> (p v q) The column p --> (p v q) should be true because tautology statements must always be true

Which statement best describes combinatorics? Combinatorics is an idea that mathematicians have yet to prove. Combinatorics is the study of shapes. Combinatorics is a fancy name for the study of counting. Combinatorics is theoretical and can't be applied to actual probabilities

Combinatorics is a fancy name for the study of counting.

Which statement is false? A universal set is a subset of itself. Every universal set has the same number of elements. A subset cannot have more elements than the set from which it is created. The elements of a universal set are based on the context of a problem.

Every universal set has the same number of elements.

If he eats a hamburger, then he will eat two bags of fries. Which of the following represents the hypothesis in the above conditional statement? If he eats a hamburger, then he will eat two bags of fries. Hamburger He will eat two bags of fries. He eats a hamburger. Then he will eat two bags of fries

He eats a hamburger. hypothesis is the question (if)

Which of these is the logical contrapositive to the following statement? If killing in any sense is wrong, then murder is wrong. If killing in any sense is wrong, then murder is not wrong. If murder is not wrong, then killing in any sense is not wrong. If murder is wrong, then killing in any sense is wrong. If killing in any sense is not wrong, then murder is not wrong.

If murder is not wrong, then killing in any sense is not wrong.

Evaluate the following sum: 4 Σ (3n-4) n = 2 24 8 6 9 15

Range is n=4 - 2 Plug in values 2-4 in 3 and add together

Which is the correct definition of a disjunction? It is when two statements are connected with an 'AND'. The resulting compound statement can be labeled as true when just one of the statements is true. It is when two statements are connected with an 'OR'. The combined compound statement can be labeled as true when just one of the statements is true. It is when two statements are connected with a 'NOT''. The resulting compound statement can be labeled as true when just one of the statements is true. It is when two statements are connected with an 'OR''; only when both statements are true is the resulting compound statement true. It is when two statements are connected with an 'AND''; only when both statements are true is the resulting compound statement true.

It is when two statements are connected with an 'OR'. The combined compound statement can be labeled as true when just one of the statements is true. Conjunction use AND disjunctions use OR

Suppose you are at a small get-together at a friend's house, and there are 13 people there, including yourself. When it comes to birthdays of the people at the party and the pigeonhole principle, which of the following statements must be true? It must be true that at least two people share the same birthday month. It must be true that at least two people were born at the same hour of day. None of these statements must be true. It must be true that at least two people have the same birthday.

It must be true that at least two people share the same birthday month.

Which of the following is NOT true about a bijection? it includes all possible outputs of a given function. It is both an injection and a surjection. It will be graphed in the Cartesian plane. The sizes of the domain and codomain must be equal.

It will be graphed in the Cartesian plane.

Select the tautology from the statements below. My name is Andrew. If it rains on Friday, then I will go to the movies. I have a cat or I have a goldfish. It will either rain or not rain.

It will either rain or not rain.

If Jimmy does his chores, then Jimmy will get a big scoop of chocolate ice cream. Which of the following represents the conclusion in the above conditional statement? A big scoop of chocolate ice cream Jimmy does his chores. If Jimmy does his chores Jimmy will get a big scoop of chocolate ice cream. If Jimmy does his chores, then Jimmy will get a big scoop of chocolate ice cream.

Jimmy will get a big scoop of chocolate ice cream.

Which of the following statements is FALSE for the Cartesian product of two sets A and B? The Cartesian product results in a set of ordered pairs of elements of sets A and B. Only one element of set A relates to an element of set B. Every element of set A relates to every other element of set B. The Cartesian product can be represented as a matrix.

Only one element of set A relates to an element of set B.

What is the first step of mathematical induction? Assume that everything is true. Prove n = k + 1 is true. Prove n = k is true. Assume that the case n = 1 is true. Prove the first case, usually n = 1, is true.

Prove the first case, usually n = 1, is true.

Which of these is the first step in mathematical induction? Show that if the statement is true for the first k elements, then it is true for the (k+1)st case. Prove that the problem you are working on is the base to all proofs. Prove the statement is true for the first element in the set. None of these are correct.

Prove the statement is true for the first element in the set.

In mathematical induction, what do you need to do after assuming n = k is true in the second step to prove the statement is true? Show that both sides of the statement equal each other. Show that both sides equal n = k + 1. Show that n = 1. Show that side a is less than side b.

Show that both sides of the statement equal each other.

Which of the following is the induction step in mathematical induction? Show that the statement is true for the first few elements in the set. None of these are correct. Show that your math problem is different from all other math problems. Show that if the statement is true for the first k elements, then it is true for the (k+1)st element in the set.

Show that if the statement is true for the first k elements, then it is true for the (k+1)st element in the set.

The converse of a logical statement is found by doing what? Negating both the hypothesis and conclusion. Negating the hypothesis and conclusion, then switching them. Negating the conclusion, but not the hypothesis. Switching the hypothesis and the conclusion.

Switching the hypothesis and the conclusion. Converse = Switch/Flip Inverse = Opposite or negate

In a mathematical induction, if n = 1 is true, what else do you have to assume to prove the statement true? The case n = 0 is true. The case n = k + 1 is true. The case n = 5 is true. The case n = k is true.

The case n = k + 1 is true.

Perform the first step of mathematical induction for the mathematical statement n + 1 > n. The case n = 0 is true. The case n = k + 1 is true. The case n = 5 is true. The case n = k is true.

The case n = k is true.

Let the universal set U be the set of Mr. Salada's 5th grade class of 17 boys and 13 girls. Let set A be the set of all the girls in Mr. Salada's class, and let set B be the set of all the boys in Mr. Salada's class. Which set has the most elements? A The complement of A The complement of U The complement of B

The complement of A

Which of these is always logically equivalent to the inverse? The contrapositive The counterexample The converse The conditional

The converse converse = inverse

In the formula for calculating an arithmetic series, what does the following term represent? a1 The difference between each term in the series The last term in the series None of the answers are correct. The first term in the series

The first term in the series

A universal set is defined as the set of natural numbers N. Therefore, U = N. Which of the following properly states in words subset A shown in the image. A = { x Є N | x < 50} The set of all natural numbers x such that x is less than 50. The set of all numbers x such that x is less than 50. The set of all natural numbers x. The set of all natural numbers x such that x is greater than 50.

The set of all natural numbers x such that x is less than 50.

Why are inverses important when it comes to isomorphisms? They allow you to graph the isomorphic equation so it mirrors the original function They allow you to find the answer quickly They allow you to guess the right answer by knowing the previous answer They allow you to find the other value regardless of which value you are given

They allow you to find the other value regardless of which value you are given

A statement can be determined to be which of the following through a direct proof? Never true Only true Only false True or false Never false

True or False Direct proof relates to conditional statements that can be proven true or false

Let the universal set U be all the letters of the English alphabet. What is the complement of the empty set? (Note: the empty set is a subset of every set.) U Ø - U {a, b, c, d} Ø

U

Let the set A = {2, 4, 6, 8} Let set B = {1, 3, 5, 7} Find. A U B

U = union meaning combine the terms from A and B ∩ = intersection meaning matching/common terms from each set

Which rule represents the nth term in the sequence 9, 16, 23, 30...? an = 7n + 9 an = 7n + 2 an = 2n + 7 an = 9n + 2 an = 9n + 7

an = 7n +2 In this problem there is no reference to where the number begin. So we have to solve for n0. n0 = d (difference) - n (pattern) +7 is the pattern between numbers first # - pattern between # 9-7 = 2

Suppose we wanted to use mathematical induction to prove that for each natural number n, 1 + 5 + 9 + ... + (4n - 3) = n(2n - 1). What would we show in the base step for n = 1 and n = 2? We would show that if the statement is true for the first k elements, then it is true for the (k + 1)st element. None of these are correct. We would show that 2 - 1 = 1. We would show that the statement was true for n = 1 and for n = 2 by plugging 1 and 2 into our formula separately, and making sure they both make a true statement.

We would show that the statement was true for n = 1 and for n = 2 by plugging 1 and 2 into our formula separately, and making sure they both make a true statement.

If a geometric series begins with the following term, what would the next term be? a1 2a1 + r a2 * r a2 + r a1 + r a1 * r

a1 * r

Choose the equation below that represents the rule for the nth term of the following geometric sequence: 128, 64, 32, 16, 8, ... an = 128(1/2)n-1 an = 1/2 (128)n-1 an = 128 (1/2)n an = 128(2)n-1

an = 128 (1/2)n-1

Which equation below represents a geometric sequence? an = -7n +1 an = n^2 +2n -1 an = 5/n an = 6 (n-1)^2 an = 4(1/2)n -1

an = 4(1/2)n -1

Which of the following functions is NOT an injection going from the set of real numbers to the set of real numbers? f(x) = 3x f(x) = x^2 f(x) = 2x / 3 f(x) = x + 1

f(x) = x^2

Which of the following words means that a circle cannot be flipped over when determining the number of different possible arrangements of items? fickle free friendly fixed

fixed

What is the rule for the nth term of the sequence with a7 = 53 and a13 = 101? an = 8n - 3 an = 8n + 3 an = -3n + 8 an = 8n + 5 an = 5n + 8

subtract a7-a13 101-53 = 48 Six places from a7 to a13 a7-a13 = 6 48/6 = 8 Counting backwards from a7 to a0 = 7 n1=53 - (space from n7 to n0 = 7 * d =8) = - 3 an = 8n - 3

Which of the following is an isomorphism? log ( x * y ) = log x / log y tan x = sin x / cos x x + 1 = x / 2 2x + y = 3y * x

tan x = sin x / cos x gives same result on both sides and doesn't change the original function

If a universal set is {1, 2, 3, 4, 5, 6, 7} and set C equals {1, 2, 3}, What is the complement of the complement of C? {1, 2, 3} Ø {4, 5, 6, 7} {1, 2, 3, 4, 5, 6, 7}

{1, 2, 3}, *complement of the complement*

If U = {11, 12, 13, 14, 15, 16, 17, 18, 19, 20} and A = {12, 13, 14, 20}, what is the complement of A? {11, 15, 16, 17, 18, 19} {15, 16, 17, 18, 19} {11, 12, 13, 14, 15, 16, 17, 18, 19, 20} {12, 13, 14, 20}

{11, 15, 16, 17, 18, 19} complement = items in the universal setting missing from set A

Which of these is an infinite sequence of all the non-zero even numbers beginning at number 2? {2, 20, 200, 2000, . . .} {2, 4, 8, 16, . . .} {2, 6, 10, 14, . . .} {2, 4, 6, 8, 10} {2, 4, 6, 8, . . .}

{2, 4, 6, 8, . . .} single digits only

U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Which of the following is not a subset of the universal set? {1, 6, 9} U {8, 9, 10} {1, 2, 3}

{8, 9, 10}

What is the correct way of denoting (or writing) the cardinality of set Q? Q = {orange, green, pink, red, black} c(Q) = 5 None of these. Q = 5 n|Q|=5 |Q| = 5

|Q| = 5