Let P(x), Q(x), and R(x) be the following predicates: P(x):x2 7x+10=0 Q(x):x2 2x

Question

Let P(x), Q(x), and R(x) be the following predicates:

P(x):x2 7x+10=0 Q(x):x2 2x3=0 R(x) : x < 0

(a) Determine the truth or falsity of the following statements, where the universe is all integers. Briefly explain your answers in precise English.

i. x, (P (x) R(x)) ii. x, (Q(x) R(x)) iii. x, (P (x) R(x))

Solution: First, note that x2 7x + 10 = (x 2)(x 5) and therefore the truth set of P(x) is {2,5}, and x2 2x3 = (x+1)(x3) and therefore the truth set of Q(x) is {1, 3}. The truth set of R(x) is the set of negative integers. The answers are: T F T

(b) Find the answers to part (a) when the universe consists of all positive integers. Solution: T F T

(c) Find the answers to part (a) when the universe contains only the integers 2 and 5. Solution: T T F

why is statement iii ture for question a, and becomes false in question c. Also why is statement ii is true for question c

Explanation / Answer

I think so answer given is wrong because in third universe is just two integers so and for both P(x) is true therefore iii has to be true for c part

Statement second is false because universe of integers is false for the statement

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