Proof Maths Help

Question

For the following claimed logical equivalence, determine whether it is true or false. Give a proof if it is true, and give a counterexample if it is false.

∀xP(x)∨∀xQ(x)≡∀x(P(x)∨Q(x))

Answer 1

Let

T = true

F = false

tval(x) = truth value of x

LHS = Left hand side of the equivalence

RHS = Right hand side of the equivalence

Universal quantifier is not distributive over disjunction. Therefore ∀xP(x)∨∀xQ(x)≡∀x(P(x)∨Q(x)) is false.

Counterexample:

Let Domain = ℝ

P(x) = x is rational

Q(x) = x is irrational

tval(LHS) = F but tval(RHS) =T

The statements "All real numbers are rational." and "All real numbers are irrational" are both false. That makes the left-hand side of the equivalence false as well.

However, if you say all real numbers are either rational or irrational, that makes the statement true because that is the very definition of real numbers.