The universal quantification of predicate \(P(x)\) over domain \(U\) is the statement “\(P(x)\) for all values of \(x\) in the domain \(U\)” and is written \(\forall x P(x)\) or \(\forall x \in U ~P(x)\). When the domain is finite, universal quantification over the domain is equivalent to iterated conjunction (ands). The existential quantification of predicate \(P(x)\) over domain \(U\) is the statement “There exists an element \(x\) in the domain \(U\) such that \(P(x)\)” and is written \(\exists x P(x)\) for \(\exists x \in U ~P(x)\). When the domain is finite, existential quantification over the domain is equivalent to iterated disjunction (ors). An element for which \(P(x) = F\) is called a counterexample of \(\forall x P(x)\). An element for which \(P(x) = T\) is called a witness of \(\exists x P(x)\).