Discrete Mathematics - Ch 2

For i = 1, 2, 3, .., define Ai = {0, 1, 2, 3, .., (10i − 1)}Ai = {0, 1, 2, 3, .., (10i - 1)}

0 ∈ ∩i=1∞ Ai0 ∈ ∩i=1∞ Ai ∩i=1∞A i = A1

Which of the following is the correct first four terms of the geometric progression with initial term 3 and common ratio 1/2?

3, 3/2, 3/4, 3/8

Match the set identity in the left-hand column with its name in the right-hand column.

A ∪ ∅ = A Identity law A = A Complementation law A ∪ (B∪C) = (A∪B)∪ C Associative law A ∩ B = B ∩ A Commutative law A ∩ ∅ = ∅ Domination law A∩BA∩B = AA ∪ BB De Morgan's law A ∪ (A∩B)A∩B = A Absorption law A ∪ A = A Idempotent law A ∩ (B∪C)B∪C = (A∩B)A∩B ∪ (A∩C)A∩C Distributive law A ∪ AA = U Complement law

Let the universal set U be the set of all students at your school. Let E be the set of students at your school majoring in engineering. Select all of the students from the list below who are members of E^-.

Vijay, who is majoring in physics and in mathematics Jermaine, who is majoring in mathematics only Scarlette, who is majoring in business only

Order the statements below, from top to bottom, to form one half of a proof that (A∪B)^- = A^- ∩ B^-

We will show Suppose Using Applying By definition Consequently

Which of the following begins the two main parts of a direct proof of A∪(B∩C)A∪B∩C = (A∪B)A∪B ∩ (A∪C)A∪C

We will show that A∪B∩C ⊆ A∪B ∩ A∪C. We will show that A∪B ∩ A∪C ⊆ A∪B∩C.

Which of the following intervals contains both the integers -1 and 3?

[-1, 5) [-1, 3] (-3, 3]

Which of the following functions from Z → R are partial functions, rather than total functions

f(n) = 1/(n^2 - 1) f(n) = √n^2 - 1

Match each set identity in the left-hand column with the corresponding logical equivalence in the right-hand column

p ∨ p ≡ p A ∪ A = A p ∧ p ≡ p A ∩ A = A p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r) matches Choice, A∪(B∩C)=(A∪B)∩(A∪C) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r) matches Choice, A∩(B∪C)=(A∩B)∪(A∩C) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) p ∨ ¬p ≡ Tp ∨ ¬p ≡ T matches Choice, A∪A=U A ∪ A = UA ∪ A = U p ∧ ¬p ≡ Fp ∧ ¬p ≡ F matches Choice, A∩A=∅ A ∩ A = ∅

Let A be the set of words containing at least two "o"s, and B be the set of words containing at least one letter "z". Select all of the following that are members of A ∩ B

zoo oozing zoology

Match the two sets on the left with their union on the right.

{a}, ∅ {a} {a}, {∅} {a, ∅} ∅, {∅} {∅} {{a}}, {∅} {{a}, ∅}

Match the logic based set builder notation on the left with the set notation on the right.

{x | x ∈ A ∨ x ∈ B} matches Choice, A∪B A ∪ B {x | x ∈ A ∧ x ∈ B} matches Choice, A∩B A ∩ B {x | x ∈ A ∧ x ∉ B} matches Choice, A-B A - B {x | x ∈ B ∧ x ∉ A} matches Choice, B-A B - A {x | x ∈ U ∧ x ∉ A} matches Choice, A A

Which of the following are subsets of {∅, {∅}}?

{{∅}} ∅ {∅} {∅, {∅}}

Which of the following sets is the power set of {a, b, {c, d}}

{∅, {a}, {b},{{c, d}}, {a, b}, {a, {c, d}}, {b, {c, d}},{a, b, {c, d}}} {{a, b, {c, d}}, {a, b},{a, {c, d}},{b, {c, d}}, ∅, {a}, {b}, {{c, d}}}

Which of the following sets are finite?

∅ {Z, N, C, Q} {Z}

Which of the following gives the correct value of ∑200 k=100 k3

(200^2 ⋅ 201^2 - 99^2 ⋅ 100^2)/4

Which of the following are true statements about the Cartesian product of {a, b} and {1, 2, 3}?

(b, 1) is an element of the Cartesian product. (a, 3) is an element of the Cartesian product. The size of the Cartesian product is 6.

Order the statements below, from top to bottom, to form one half of a proof that A∪B = A ∩ B

1) We will show that A^- U^- B^- C_ A^- ∩ B^- 2) SUppose 3) Using 4) Applying 5) By definition 6) Consequently

Suppose a person deposits $100,000 in a savings account yielding 2% a year with interest compounded annually. Determine how much will be in the account after 10 years by calculating the value of P10, where Pn is the amount in the account after n years

121,899.44

Let U = {1,2,3,4,5,6,7,8} be a universal set, and the ordering of elements of U has the elements in increasing order. A set A has bit string representation 10110101. Select all of the following that are members of A

4 8 6 3 1

Let U = {1,2,3,4,5,6,7,8,9} be the universal set, and let A = {2,3,4,7,8}. Select all of the following that are members of A

6 1 9 5

Match the statement on the left with the task on the right needed to prove the statement.

A ⊆ B Show that if x ∈ A, then x ∈ B. A ⊈ B Find a single x ∈ A, such that x ∉ B. A = B Show that A ⊆ B and B ⊆ A. A = ∅ Show that A ⊆ ∅. A ≠ ∅ Find a single x ∈ A.

Select all that are set identities

A∪(B∪C) = (A∪B)∪C A ∩ ∅ = ∅ A ∩ B = B ∩ A A∩(B∩C) = (A∩B)∩C A ∪ ∅ = A A∪(A∩B)=A A ∪ U = U A ∪ A = A (A∩B)^- = A^- ∪ B^- A^-^- = A A ∩ A^- = ∅ A∪(B∩C) = (A∪B) ∩ (A∪C) A ∪ B = B ∪ A A∩(B∪C) = (A∩B) ∪ (A∩C) A ∪ A^- = U (A∪B)^- = A^- ∩ B^- A∩(A∪B) = A A ∩ A = A

Consider two multisets of tools, BikeBag = {3 ⋅ screw driver, 4 ⋅ wrench, 2 ⋅ patch, 1 ⋅ pump} and GarageShop = {7 ⋅ screw driver, 22 ⋅ wrench, 1 ⋅ pump}. Match each set described on the left with its value on the right

BikeBag ⊔ GarageShop matches Choice, {7 ⋅screw driver, 22 ⋅wrench, 2 ⋅patch, 1 ⋅pump} {7 ⋅ screw driver, 22 ⋅ wrench, 2 ⋅ patch, 1 ⋅ pump} BikeBag ⊓ GarageShop matches Choice, {3 ⋅screw driver, 4 ⋅wrench, 1 ⋅pump} {3 ⋅ screw driver, 4 ⋅ wrench, 1 ⋅ pump} BikeBag + GarageShop matches Choice, {10 ⋅screw driver,26 ⋅wrench, 2 ⋅patch, 2 ⋅pump} {10 ⋅ screw driver,26 ⋅ wrench, 2 ⋅ patch, 2 ⋅ pump} BikeBag - GarageShop matches Choice, {2 ⋅patch} {2 ⋅ patch}

Compute the floor or ceiling, as indicated by the notation, of each of the following

Blank 1: 1 Blank 2: 0

Consider a function G that maps students in a class to grades. The domain is the set of students in the class: {Yu, Kirstein, Alshahrani, Yarra, Mitchell, Feldman}.The codomain is the set of grades: {A, B, C, D, E, F}. Select all of the functions below that are bijections.

G(Yu) = F, G(Kirstein) = E, G(Alshahrani) = D, G(Yarra) = C, G(Mitchell) = B, G(Feldman) = A G(Yu) = A, G(Kirstein) = B, G(Alshahrani) = C, G(Yarra) = D, G(Mitchell) = E, G(Feldman) = F

Suppose that f: A → B. Match the property on the left with the corresponding method for showing that the property holds on the right

Injective -> Show that if f(x) = f(y) for x, y ∈ A, then x = y. Not injective -> Find particular elements x, y ∈ A with x ≠ y and f(x) = f(y). Surjective -> Consider an arbitrary element y ∈ B and find an element x ∈ A such that f(x) = y. Not surjective -> Find a particular y ∈ B such that f(x) ≠ y for all x ∈ A

Let E be the set of students at your school majoring in engineering and let B be the set of students at your school majoring in business. Select all of the students from the list below who would be members of E - B

Karson, who is majoring in engineering only Avi, who is majoring in both computer science and engineering

Let E be the set of students at your school majoring in engineering, and let B be the set of students at your school majoring in business. Select all of the students from the list below who are members of E ∩ B.

Krishna, who is majoring in computer science, engineering, and business Phoebe, who is majoring in both engineering and business

Suppose P and Q are the multisets {3 ⋅ a, 4 ⋅ b} and {1 ⋅ a, 3 ⋅ b}, respectively.

P ⊔ Q {3 ⋅ a, 4 ⋅ b} P ⊓ Q {1 ⋅ a, 3 ⋅ b} P + Q {4 ⋅ a, 7 ⋅ b} P - Q {2 ⋅ a, 1 ⋅ b}

Let E be the set of students at your school majoring in engineering and let B be the set of students at your school majoring in business. Select all of the students from the list below who would be members of E ∪ B

Scarlette, who is majoring in business Dan, who is majoring in computer science and in business Phoebe, who is majoring in both engineering and business Karson, who is majoring in engineering

Let V be the set of all restaurants in your town that serve vegan food, G the set of all restaurants that serve gluten-free food, and K the set of all restaurants in your town that serves food certified as kosher. Select all of the following that accurately describe the set V ∩ G ∩ K

The set of all restaurants in your town that serve gluten-free food, certified kosher food, and vegan food. The set of all restaurants in your town that serve vegan food and gluten-free food and certified kosher.

Given the sequence {an} with initial terms -2, -1, 1, 5, 13, 29, 61, which approach can help you find a simple formula for {an}

The terms differ from those of the sequence {2n} by an amount that can be expressed with a simple formula.

Which of the following begins the two main parts of a direct proof of A∪(B∩C)A∪B∩C = (A∪B)A∪B ∩ (A∪C)

We will show that A∪(B∩C) ⊆ (A∪B) ∩ (A∪C). We will show that (A∪B) ∩ (A∪C) ⊆ A∪(B∩C).

Let A = {a,e,i,o,u} and B = {c,d,e,i,r,s,t} Select all of the following that are members of A - B

a o u

Match the first five terms on the right with the sequence defined by the recurrence relation and initial conditions on the left

an = a2n−1an-12, a1 = 1, a2 = -1 -> 1, -1, 1, 1, 1 an = 2an-1 - an-2, a1 = 1, a2 = -1 -> 1, -1, -3, -5, -7 an = 4an-1 - 2n, a1 = 1, -> 1, 0 -8, -48, -224 an = a2n−1 − a2n−2an-12 - an-22, a1 = 1, a2 = -1 -> 1, -1, 0, -1, 1

Match each recurrence relation on the left with the value of its third term as on the right

an = an−1 + 2an−2 , a1 = 3,a2 = 2, 8 an = an−1 . an−2,a1 = 2, a2 = 7 14 an = an−12, a1 = 2 16 an=nan−1, a1=2 12 an=2an−1+1, a1=3 15

Match the sequence {a_n} on the left with the value of a4 on the right

an = n + 3 7 an = 2n + 1 9 an = 3^(n - 3) 3 an = 2^n - 8 8 an = n^2 - n 12

Let A and B be nonempty sets. If f is a function from A to B, A is the ______ of the function and B is the ______ of the function. If f(a) = b, b is the image of a and a is the ______ of b. The ______ of f is the set of all images of elements of A

domain codomain preimage range, image

Match the function from R to R with its inverse.

f(x) = 1 - x -> f -1(x) = 1 - x f(x) = x^2 + 1 -> Not invertible f(x) = 1 + x -> f -1(x) = x - 1 f(x) = x^3 -> f ^-1(x) = x⎯⎯√3 f(x) = ^3√x -> f^ -1(x) = x^3

Which of the following functions from R to R are invertible?

f(x) = 2x + 7 f(x) = x5 - 1 f(x) = x⎯⎯√5

Match the functions f(x), g(x) on the left with composition (f∘g)(x) on the right

f(x) = 4x + 3, g(x) = 3x + 4 12x+19 f(x) = 2x + 1, g(x) = 6x + 2 12x + 5 f(x) = 3x + 4, g(x) = 4x + 3 12x + 13 f(x) = 6x + 2, g(x) = 2x + 1 12x + 8

The formulas on the left each define a function with domain and codomain equal to Z. Match each function with its range from the right

f(x) = x^3 The perfect cubes g(x) = |x| The nonnegative integers h(x) = 2x The even numbers i(x) = x + 1 All integers

Let A be the set of words containing at least one "j" and B be the set of words containing at least one "k." Select all of the words that are members of A ∪ B.

jail joke jack kangaroo

Which of the following are correct ways to describe the set of nonnegative integers less than or equal to 100 that are perfect cubes?

{0, 1, 8, 27, 64} { x | x = y3 where y is a nonnegative integer not exceeding 4} { x | x is an integer, 0≤ x ≤ 100 and x is a perfect cube}

Match the sets on the left with a true statement about the Cartesian product of those sets on the right.

{1, 2} and {3, 4} matches Choice, Its cardinality is 4. Its cardinality is 4. {1, 2, 3, 4} and {3, 4, 5, 6} matches Choice, (4, 3) is a member. (4, 3) is a member. {4, 5, 6, 7} and {4, 5, 6, 7} matches Choice, (5, 5) is a member. (5, 5) is a member. {a, e, i, o, u} and {b, g, t, d} matches Choice, Its cardinality is 20. Its cardinality is 20. {1, 2, 3} and {1, 2, 4} matches Choice, (2, 2) is a member. (2, 2) is a member.

Match a set on the left with an alternate way of describing it on the right

{2, 4, 6, 8, 10} matches Choice, {x | x is a positive even integer not exceeding 10} {x | x is a positive even integer not exceeding 10} {1, 2, 4, 8} matches Choice, {x | x is a positive integer which is a power of 2, not exceeding 10} {x | x is a positive integer which is a power of 2, not exceeding 10} {2, 6, 10} matches Choice, {x | x is a positive even integer not divisible by 4, and not exceeding 10} {x | x is a positive even integer not divisible by 4, and not exceeding 10} {0, 2, 4, 6, 8} matches Choice, {x | x is a nonnegative even integer, less than 10} {x | x is a nonnegative even integer, less than 10}

Match the set on the left with its size on the right.

∅ matches Choice, 0 0 {∅,{∅}} matches Choice, 2 2 {∅,∅} matches Choice, 1 1 The set of letters in the English alphabet. matches Choice, 26 26 {a, b, S}, where S is the set of letters in the English alphabet. matches Choice, 3 3 Z matches Choice, infinite infinite

Match the set on the left with its power set on the right

∅ {∅} {∅} {{∅},∅} {{∅,{∅}}} {∅,{∅,{∅}}} {∅,{∅}} {∅,{∅},{{∅}}, {∅,{∅}}}

Match the summation on the left with its value on the right

∑6j=1 (j2−2) = 79 ∑4j=2 (j3−j4) = -254 ∑4j =1 ∑3 i=1 (ij + 1) = 72 ∑7k=4 k2 = 126 ∑ 3j=0 3j = 18

Which of the following statements are correct for all integers n and real numbers x?

⌈x⌉ = nx = n if and only if x ≤ n < x + 1x ≤ n < x + 1. ⌊x + n⌋ = ⌊x⌋ + n