Discussion
Which one of the following well-formed formulae in predicate calculus is NOT valid?
Answer: D
The formula that is NOT valid is ∀x (p(x) ∨ q(x)) ⇒ (∀x p(x) ∨ ∀x q(x)).
Counterexample: let the domain be the integers, \(p(x)\) = “x is even” and \(q(x)\) = “x is odd”. Every integer is even or odd, so the left side is true, but neither “all integers are even” nor “all are odd” is true.
The others are valid. (A): if \(\forall x\,p(x)\) fails then \(\exists x\neg p(x)\) holds, and if it holds then the premise gives \(\forall x\,q(x)\). (B): an existential distributes over disjunction, so the two sides are equivalent. (C): a witness of \(p \land q\) is a witness for each separately.
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