Quantifiers are used in mathematical logic to express the extent to which a statement is true or false over a specified domain. There are two types of quantifiers: universal quantifiers and existential quantifiers.
Universal quantifiers are used to indicate that a statement holds for every element in a set or domain. The universal quantifier is denoted by the symbol "∀", which is read as "for all". For example, the statement "For all positive integers n, n + 1 > n" can be written as ∀n ∈ Z⁺, n + 1 > n, where Z⁺ is the set of positive integers.
Existential quantifiers are used to indicate that there exists at least one element in a set or domain that satisfies a given condition. The existential quantifier is denoted by the symbol "∃", which is read as "there exists". For example, the statement "There exists a positive integer n such that n² = 4" can be written as ∃n ∈ Z⁺, n² = 4.
De Morgan's law for quantifiers states that the negation of a quantified statement is the same as quantifying the negation of the statement. For example, the negation of "For all x, P(x)" is "There exists an x such that ¬P(x)", and the negation of "There exists an x such that P(x)" is "For all x, ¬P(x)". This can be written symbolically as:
¬(∀x, P(x)) ⇔ ∃x, ¬P(x)
¬(∃x, P(x)) ⇔ ∀x, ¬P(x)
De Morgan's law for quantifiers can be useful for simplifying logical expressions and proving certain statements in mathematical logic.
