Discrete Mathematics Exam 1 (Ch. 1 and 2)

The sequence whose nth term is n! − 2n where n ≥ 1.

-1,-2,-2,8,88,656,4192,40064,362368,3627776

The sequence that lists each positive integer three times, in an increasing order.

1,1,1,2,2,2,3,3,3,4

The sequence that lists the odd positive integers in an increasing order, listing each odd integer twice.

1,1,3,3,5,5,7,7,9,9

Let A = {a, b, c, d, e} and B = {a, b, c, d, e, f, g, h}. Match the given sets in the left to the sets in the right. 1. A − B 2. A ∩ B 3. A ∪ B 4. B − A

1. Ø 2. {a, b, c, d, e, f, g, h} 3. {a, b, c, d, e} 4. {f, g, h}

Let A = {1, 2, 3, 4, 5} and B = {0, 3, 6}. Match the given sets on the left to the sets on the right. 1. A ∩ B 2. B − A 3. A ∪ B 4. A − B

1.{0, 1, 2, 3, 4, 5, 6} 2.{0, 6} 3.{3} 4.{1, 2, 4, 5}

The sequence that begins with 2 and in which each successive term is 3 more than the preceding term.

2,5,8,11,14,17,20,23,26,29

Identify the correct statements for the given functions from the set {a, b, c, d} to itself. (Check all that apply.)

f(a) = b, f(b) = a, f(c) = c, f(d) = d is a one-to-one function, as each element is an image of exactly one element. f(a) = b, f(b) = b, f(c) = d, f(d) = c is a one-to-one function, as each element is mapped to only one element. f(a) = b, f(b) = b, f(c) = d, f(d) = c is not a one-to-one function, as b is an image of more than one element. f(a) = d, f(b) = b, f(c) = c, f(d) = d is a one-to-one function, as each element is mapped to only one element. f(a) = d, f(b) = b, f(c) = c, f(d) = d is not a one-to-one function, as d is an image of more than one element.

The compound proposition for the statement "It is below freezing but not snowing" is _____.

p ∧ ¬q

Among the following pairs of sets, identify the ones that are equal. (Check all that apply.)

{ {1} }, {1, {1} } {1, 2}, { {1}, {2} }

Find the power set of each of the following sets, where a and b are distinct elements. {b}

{ Ø, {b} }

Match the sets on the right column to the sets on the left column by clicking and dragging them. {x | x is a real number such that x^2=1} {x | x is a positive integer less than 12} {x | x is the square of an integer and x < 100} {x | x is an integer such that x^2=2}

{1,-1} {1,2,3,4,5,6,7,8,9,10,11} {0,1,4,9,16,25,36,49,64,81} Ø

Use De Morgan's laws to find the negation of the given statement. Yoshiko knows Java and calculus.

Yoshiko does not know Java or does not know calculus.

For the sequence an = ⌊n/2⌋⌊n/2⌋ , the values of the first four terms of the sequence are

a0 = 0, a1 = 0, a2 = 1, and a3 = 1

For the sequence an = (n + 1)(n + 1), the values of the first four terms of the sequence are

a0 = 1, a1 = 4, a2 = 27, and a3 = 256

For the sequence an = 2n + 1, identify the values of a0 = _____, a1 = _____, a2 = _____, and a3 = _____.

a0 = 2, a1 = 3, a2 = 5, and a3 = 9

For the sequence an = −2an − 1 and a0 = −1, the values of the first six terms are

a0 =-1 : a1 =2 : a2 =-4 : a3 =8 : a4 =-16 : a5 =32

For the sequence an = 3a2n - 1 and a0 = 1, the values of the first six terms are

a0 =1 : a1 =3 : a2 =27 : a3 =2187 : a4 =14348907 : a5 =617673396283947

For the sequence an = an - 1 - an - 2, a0 = 2, and a1 = -1, the values of the first six terms are

a0 =2 : a1 =-1 : a2 =-3 : a3 =-2 : a4 =1 : a5 =3

Let R(x) is "x is a rabbit" and H(x) is "x hops," and the domain consists of all animals. Translate the statement ∀∀ x(R(x) Λ H(x)) into English.

Every animal is a rabbit and hops.

∀x∃y(x < y) where x and y are real numbers.

For every real number x there exists a real number y such that x is less than y.

p: "Swimming at the New Jersey shore is allowed." q: "Sharks have been spotted near the shore."

Swimming at the New Jersey shore is allowed, and sharks have been spotted near the shore.

Determine whether ¬(¬p) and p are logically equivalent or not by completing the given truth table.

T F

Complete the truth table given below to verify the first De Morgan law ¬(p ∧∧ q) ≡ ¬p ∨∨ ¬q.

T F F F T T

Let Q(x, y) denote the statement "x is the capital of y." Determine the truth value of Q(Massachusetts, Boston).

The truth value of Q(Massachusetts, Boston) is false

p: "Swimming at the New Jersey shore is allowed." q: "Sharks have been spotted near the shore." Check if the proposition "Sharks have not been spotted near the shore" is the expression for ¬q.

True

p: It is below freezing. q: It is snowing. The compound proposition for the statement "It is below freezing and snowing" is p ∧ q.

True