Negation of the proposition \(∃𝑥𝐻(𝑥)\) is

Answer: ∀x ¬H(x).

Reason: By the quantifier negation rule, not-exists becomes for-all-not: ¬(∃x H(x)) ≡ ∀x ¬H(x). Intuitively, saying "it is not the case that some x has property H" means "every x does not have property H."

• Common incorrect alternative: ∃x ¬H(x) states that there exists at least one x without H, which is not equivalent to saying no x has H.

• Example to see the difference: if H(1) = true and H(2) = false, then ∃x H(x) is true and its negation is false, but ∃x ¬H(x) is true. So ∃x ¬H(x) does not capture the negation of ∃x H(x).