# Propositional function problems

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1. Aug 10, 2017

### Mezza

1. Suppose P(x) and Q(x) are propositional functions and D is their domain.
Let A = {x ∈ D: P(x) is true}, B = {x ∈ D: Q(x) is true}

(a) Give an example for a domain D and functions P(x) and Q(x) such that A∩B = {}
(b) Give an example for a domain D and functions P(x) and Q(x) such that A ⊆ B but A ≠ B.
(c) Given that x ∈ A - B, what is the truth value of Q(x)?
(d) Given that x ∈ A - B, what is the truth value of P(x) ∨ ¬Q(x)?

My Attempt.

(a) Let P(x): x ≥ 0, Q(x): x < 0
Domain = ℝ

∴A∩B = {}

(b) Let P(x): x ≥ 2, Q(x): x ≥ 3
Domain = ℝ

∴ A ⊂ B (meaning A ⊆ B but A ≠ B)

(c) Well, we let A = {x ∈ D: P(x) is true} and we are only in part of set A with no overlap with set B.

Only P(x) is true in this part of A (no overlap with set B where Q(x) is true).

∴ Q(x) is false.

(d) For x ∈ A - B; P(x) ∨ ¬Q(x) = TRUE OR (NOT FALSE) = TRUE OR FALSE = TRUE.

I'm brand new to logic and I'd like to check my solutions for any errors and / or improvements.

Cheers.

2. Aug 15, 2017 at 1:12 PM

### Ssnow

In the point $b$ we have that $Q(x)\Rightarrow P(x)$ so $B \subset A$ and not what you said ...
Ssnow