Week 3 Quiz

The propositional variables f, h, and p represent the propositions: f: The student got an A on the final. h: The student turned in all the homework. p: The student is on academic probation Select the logical expression that represents the statement: "The student is not on academic probation and the student got an A on the final or turned in all the homework." a) ~ (p^f) V h b) (~p^f) V h c) ~p^(fVh) d) ~p^f^h

c) ~p^(fVh)

The domain of discourse are the students in a class. Define the predicates: S(x): x studied for the test A(x): x received an A on the test Select the logical expression that is equivalent to: "Someone who did not study for the test received an A on the test."

∃x(¬S(x)∧A(x))

Prove: For any odd integer n , the number 2 n 2 + 5 n + 4 is odd.

Assume: n is an odd integer. so n = 2k +1 for some integer k. then 2n^2 + 5n + 4 = 2(2k +1)^2 + 5(2k +1) + 4 = 2(4k^2+4k+1)+ (10k + 5) + 4 = (8k^2+8k+2)+ (10k + 5) + 4 = 8k^2 + 18k + 11 = 2(4k^2 + 9k + 10) + 1 where 4k^2 + 9k + 10 is an integer therefore 2n^2 + 5n + 4 is odd.

Theorem: For any real number x, if 0 ≤ x ≤ 3 , then 15 − 8 x + x 2 > 0 Which facts are assumed and which facts are proven in a proof by contrapositive of the theorem?

Assumed: 15 − 8 x + x 2 ≤ 0 Proven: x < 0 or x > 3

Theorem: For any two real numbers, x and y, if x and y are both rational then x + y is also rational. Which facts are assumed and which facts are proven in a direct proof of the theorem?

Assumed: x is rational and y is rational Proven: x + y is rational

Select the mistake that is made in the proof given below. Theorem. The product of an even integer and any other integer is even. Proof. Suppose that x is an even integer and y is an arbitrary integer. Since x is even, x = 2k for some integer k. Therefore, xy = 2m for some integer m, which means that xy is even.

Assuming facts that have not yet been proven.

Select the statement that is false. a) If 3 is a prime number, then 5 is a prime number. b) If 4 is a prime number, then 6 is a prime number. c)f 4 is a prime number, then 5 is a prime number. d) If 3 is a prime number, then 6 is a prime number.

If 3 is a prime number, then 6 is a prime number.

Write the contrapositive of: "If x≠4, then 3x≠12."

If 3x = 12, then x=4

Select the expression that is equivalent to the following statement: Among any two consecutive positive integers, there is at least one integer that is not prime.

If x is a positive integer, then x is not prime or x+1 is not prime.

Use De Morgan's law to write a statement (in English) that is logically equivalent to: "It is not true that every student got an A on the test."

There is a student who did not get an A on the test.

The domain for variable x is the set {Ann, Ben, Cam, Dave}. The table below gives the values of predicates P and Q for every element in the domain. Name P(x) Q(x) Ann F F Ben T F Cam T T Dave T T Select the statement that is false. a) ∃x (P(x) → Q(x)) b) ∃x (P(x) ∧ ¬Q(x)) c) ∃x (P(x) ∧ Q(x)) d) ∃x (¬P(x) ∧ Q(x))

d) ∃x (¬P(x) ∧ Q(x))

The propositional variables s and m represent the two propositions: s: It is sunny today. m: I will bring my umbrella. Give the logical expression that represents the statement: "Despite the fact that it is sunny today, I will bring my umbrella." Use symbols v, ^, and ~ as needed.

s^m

Select the statement that is equivalent to the statement: It is not true that x < 7

x ≥ 7

The domain for variable x is the set of all integers. Select the statement that is true.

∃ x ( x^2 < 1 )

The domain for x and y is the set of real numbers. Select the statement that is false.

∃ x ∀ y ( x + y ≥ 0 )

p = F, q = T, and r = T. Select the expression that evaluates to false.

∼ q