Discrete Math Negating Nested Quantifiers Negating Nested Quantifiers

Discrete Math: Negating Nested Quantifiers

Negating Nested Quantifiers Statements involving nested quantifiers can be negated by successively applying the rules for negating statements involving a single quantifier. Example: Express the negation of the statement ∀x∃y(xy = 1) so that no negation precedes a quantifier. Solution: By successively applying De Morgan’s laws for quantifiers in Table 2 of Section 1. 4, we can move the negation in ¬∀x∃y(xy = 1) inside all the quantifiers. We find that ¬∀x∃y(xy = 1) is equivalent to ∃x¬∃y(xy = 1), which is equivalent to ∃x∀y¬(xy = 1). Because ¬(xy = 1) can be expressed more simply as xy = 1, we conclude that our negated statement can be expressed as ∃x∀y(xy = 1). To be continued in the next slide….

Negating Nested Quantifiers Solution con’t: ∀w¬∀a∃f (P (w, f ) ∧ Q(f, a)) ≡ ∀w∃a¬∃f (P (w, f ) ∧ Q(f, a)) ≡ ∀w∃a∀f ¬(P (w, f ) ∧ Q(f, a)) ≡ ∀w∃a∀f (¬P (w, f ) ∨ ¬Q(f, a)). This last statement states “For every woman there is an airline such that for all flights, this woman has not taken that flight or that flight is not on this airline. ”

References Discrete Mathematics and Its Applications, Mc. Graw-Hill; 7 th edition (June 26, 2006). Kenneth Rosen Discrete Mathematics An Open Introduction, 2 nd edition. Oscar Levin A Short Course in Discrete Mathematics, 01 Dec 2004, Edward Bender & S. Gill Williamson