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Propositional function problems

  1. Aug 10, 2017 #1
    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. jcsd
  3. Aug 15, 2017 at 1:12 PM #2

    Ssnow

    User Avatar
    Gold Member

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