Discrete Math

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

960 a_n = a_0+nd

Which of the following is not a finite set?

A = the set of all real numbers. A set that is set that is infinite in the number of elements

Which of the following pair of sets is equal?

A = {1,2,3} B = {3,2,1 Sets with the same items in any order are still equal

Which of the following statements is FALSE? A POSET is called a lattice if it is both a join semilattice and meet semilattice. The element which is not related to any other element in a Hasse diagram is called maximal element. 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.

A relation R, over a set A, is reflexive if every element of the set is related to itself.

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.

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

A.

Which of these questions should you ask yourself to help determine if mathematical induction is a good method to prove a given statement? 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? 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. Can I prove the first few cases easily? 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.

Which of the following statements about the pigeonhole principle is 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. The pigeonhole principle is a simple yet extremely useful concept used in combinatorics. All of these statements about the pigeonhole principle are true. 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.

All of these statements about the pigeonhole principle are true.

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

An injective function

Which logical fallacy, I will be an A in math because I spent many sleepless nights studying

Appeal to emotion

Which logical fallacy, the moon is made of cheese because it looks like cheese and no one has ever eaten the moon

Appeal to ignorance

Which logical fallacy, the chocolate pie must be bad because nobody is taking it. Or Everyone in the class agrees that if x=2 and y=2, then x must equal y

Appeal to popularity

Logical fallacy when only a few choices are presented but there are more valid choices

Limited choice

Saying something is true or false based on emotion. Any reasoning not based on pure facts.

Logical fallacy

Which of the following pair of sets is equivalent?

M = {red, green, brown, blue} N = {5,6,7,8} Equivalent means same number of items, equal means same items in any order

What is the inverse of a statement?

Negative form If I am in Kansas the inverse is I am NOT in Kansas

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. Every element of set A relates to every other element of set B. Only one element of set A relates to an 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.

Evaluate f(x) = √(x), when x = 9.

Plug in x, Square root of 9 = 3

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

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

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

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

What is the correct way of denoting (or writing) the cardinality of set Q? Q = {orange, green, pink, red, black}

Q = 5 or (n)Q=5 cardinality is the amount of items in the set

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 € N | x < 50 }

Set of all natural numbers such that x is less than 50 € = is an element of all natural numbers

Let set A= {socks, shoes, pants, and shorts} Let set B= {shorts, sandals, t-shirt}. Find A n B

Shorts is the intersection because it's present in both sets A and B

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 side a is less than side b. Show that n = 1. Show that both sides of the statement equal each other. Show that both sides equal n = k + 1.

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. 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 your math problem is different from all other math problems. None of these are correct.

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.

What is the converse of a logical statement?

Switching the hypothesis and conclusion

In a mathematical induction, if n = 1 is true, what else do you have to assume to prove the statement true? Show that side a is less than side b. Show that n = 1. Show that both sides of the statement equal each other. Show that both sides equal n = k + 1.

The case n = k is true.

What is always logically equivalent to the inverse?

The converse

In the formula for calculating an arithmetic series, what does the following term represent? a^1 None of the answers are correct. The difference between each term in the series The last term in the series The first term in the series

The first term in the series

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

tan x = sin x / cos x

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

2

Perform the first step of mathematical induction for the mathematical statement n + 1 > n. 2 > 1 2 < 1 0 > 1 2 < 3 2 > 2

2 > 1

Find the sum of the following geometric series: 174,763 2,044 511 1,024 2,048

2044

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? 24 30 33 21 27

27

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? 3 2 4 1

3

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

4, -13, 1, 5, 16, ... random change from one number to the next, no pattern

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

40,320

Solve the expression 7P2 (P = permutation) 2,640 42 120 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? 518 944 720 360 472

472

Which sequence below is a geometric sequence? 0, 1, 1, 2, 3, ... 2, 3, 4, 5, 6, ... 6, 8, 11, 14, 18, ... 14, 16, 18, 20, 22, ... 5, 15, 45, 135, 405, ...

5, 15, 45, 135, 405, ...

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? 27 locker combinations 362,880 locker combinations 252 locker combinations 504 locker combinations

504 locker combinations

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

75

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

{8,9,10} because 10 is not in the universal set U

Logically fallacy when someone makes assumptions without sufficient evidence like assuming all dogs like rock and roll because one person's dog does

Hasty Generalization

Which of the following is a conditional statement

If then format, if is hypothesis, then is conclusion

What is the most specific term that describes the numbers 5, 10, 15, 20, 25, ...? Infinite sequence Infinite geometric sequence Infinite arithmetic sequence Finite sequence Finite arithmetic sequence

Infinite arithmetic sequence

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 were born at the same hour of day. It must be true that at least two people share the same birthday month. It must be true that at least two people have the same birthday. None of these statements must be true.

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 is both an injection and a surjection. it includes all possible outputs of a given function. The sizes of the domain and codomain must be equal. It will be graphed in the Cartesian plane.

It will be graphed in the Cartesian plane.

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

-237 This gives: an = -4(n - 1) + 91. Plugging in 83: a83 = -4(83 - 1) + 91 = -4(82) + 91 = -328 + 91 = -237.

If the domain of the function f(x) = x2 is all real numbers, what is the range of f? 0 to infinity 0 to -0 -infinity to 0 -1 to 1 -infinity to infinity

0 to infinity

What is the domain of √(x)? -1 to 1 0 to infinity -infinity to 0 0 to 1

0 to infinity

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? a v 3 14 19 24 29 9

19 it's the 3rd term

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

12 + 10 + 8 + 6 + 4

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? 18 120 30 60

120

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

120

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

144 Add the two terms together

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

145

Evaluate the following sum:

15

Evaluate h(x) = 5x + 1, when x=3. 16 x 6 4 3

16

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. Assume: 2 + 5 + 8 + ... + (3k - 1) + (3(k +1) - 1) = (k + 1)(3(k+1) - 1) / 2 Show: 2 + 5 + 8 + ... + (3k - 1) = k(3k - 1) / 2 None of these are correct. 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 Assume: 2 - 3 - 8 - ... - (3k + 1) = 3k / 2 Show: 2 - 3 - 8 - ... - (3(k - 1) - 1) = 3(k + 1) / 2

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? Reflexive relation Anti-symmetric relation Transitive relation Asymmetric relation

Asymmetric relation Required would be: Reflexive, ANTI-aymmetric, and transitive

If you're placing pigeons in pigeonholes, and there are more pigeons than pigeonholes, which of the following must be TRUE? All of the pigeons will need to be placed in one hole. The pigeons that have the same coloring will need to be placed together. 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.

Logic Statement/Proposition

Can only be true or false. I.e Circles Roll or Sam only eats square food

The moon is made of cheese because the moon is made of cheese

Circular reasoning

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

Combinatorics is a fancy name for the study of counting.

Switch the if with then then negate them not

Contrapositive

Statements that start with a hypothesis then use facts to support them. A conditional statement is proven

Direct proof

Which one is the correct denotation of an empty set?

E = { }

Which one is the correct demotion of an empty set?

E = {}

Given A = {1, 2, 3, T, U, B, E} Which of the following is an element of set A?

E because any element is any integer or character with set A {}

Which statement is false?

Every universal set has the same number of elements. Ex the number of presidents is different than the number of planets

Type of reasoning that uses to unrelated events to prove a point. My phone starts to ring whenever my car starts to purr

False cause

The amount of money you spend on coffees every month can be calculated as a function of the number of drinks you order every month. What are the independent and dependent variables in this function? The independent variable is the amount of money you spend on coffee per month and the dependent variable is the number of drinks you order per month. The independent variable is the number of drinks you order every month and the dependent variable is the number of months. The independent variable is the number of drinks you order per month and the dependent variable is the amount of money you spend on coffee per month. The independent variable and dependent variable cannot be determined from the given situation. The independent variable is the number of months and the dependent variable is the amount of money you spend on coffee per month.

The independent variable is the number of drinks you order per month and the dependent variable is the amount of money you spend on coffee per month. Drinks are independent because the amount of drinks won't change. Money spent is dependent because it changes if the costs of the coffee changes

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

The union is 1,2,3,4,5,6,7,8 because set A and B are combined

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 other value regardless of which value you are given 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

Negation removes the not. What is the negation of triangles are not squares

Triangles are squares

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 2 - 1 = 1. 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 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 x=1 and y=2, what can be said about z if z =xy?

Z = 1x2=2

Find the sum of the following geometric series: 3 1 6 8 4

a. 3

If a geometric series begins with the following term, what would the next term be? a^1 a2 * r a2 + r 2a1 + r a1 + r a1 * r

a1 * r

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

an = 2(3)n-1

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

an = 7n + 2

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

an = 8n - 3

The formula for a finite geometric series is: s = r1(1 -nr)(1 -r) s = r2(1 - nr)(1 -r) s = a1(1-rn)/1-r s = (a-1)(1-nr)/1-r s = an (1-ar)+(1-r)

c

Express this series using sigma notation: 5 + 6 + 7 + 8 + 9 + 10...

d.

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. 24.00 feet 2.63 feet 1.10 feet 3.43 feet 3.75 feet

d. 3.43 feet

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 + 1 f(x) = x2 f(x) = 2x / 3

f(x) = x2