- #1
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.
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.