9.14.2023
Question 1: Which of the following is the negation of the open statement $\forall x [p(x) \wedge \neg q(x)]$?
(3 points)
A. $\forall x \; [\neg p(x) \vee q(x)]$
B. $\exists x \; [\neg p(x) \vee q(x)]$
C. $\exists x \; [\neg p(x) \wedge q(x)]$
D. $\exists x \; [ p(x) \vee \neg q(x)]$
Answer
B. $\exists x \; [\neg p(x) \vee q(x)]$Question 2: Let $p(x,y)$ denote the open statement “$x$ divides $y$” where the universe for $x$ and $y$ is all positive integers and “divides” means “divides evenly”. Which of the following statements is false?
(3 points)
A. $p(3, 27)$
B. $\forall x \; p(x, 0)$
C. $\forall y \; p(1, y)$
D. $\forall x \; \forall y \; p(x, y)$
Answer
D. $\forall x \; \forall y \; p(x, y)$Question 3: Suppose you have the following open statements: $p(x): \; x^{2} - 8x + 15 = (x - 3)(x-5) = 0$ and $q(x): \; x$ is odd. Which of the following statements is false?
(3 points)
A. $\exists x \; [q(x) \rightarrow p(x)]$
B. $\exists x \; [p(x) \rightarrow q(x)]$
C. $\forall x \; [q(x) \rightarrow p(x)]$
D. $\forall x \; [\neg q(x) \rightarrow \neg p(x)]$