Discrete Math Universal Quantifier THE UNIVERSAL QUANTIFIE Many

Discrete Math: Universal Quantifier

THE UNIVERSAL QUANTIFIE • Many mathematical statements assert that a property is true for all values of a variable in a particular domain, called the domain of discourse (or the universe of discourse), often just referred to as the domain. • Such a statement is expressed using universal quantification. • The universal quantification of P (x) for a particular domain is the proposition that asserts that P (x) is true for all values of x in this domain. Note that the domain specifies the possible values of the variable x.

THE UNIVERSAL QUANTIFIE • The meaning of the universal quantification of P(x) changes when we change the domain. • The domain must always be specified when a universal quantifier is used; Ø without it, the universal quantification of a statement is not defined.

THE UNIVERSAL QUANTIFIE

THE UNIVERSAL QUANTIFIE ∀x. P(x) • When True? Ø P (x) is true for every x. • When False? Ø There is an x for which P (x) is false.

THE UNIVERSAL QUANTIFIE Example: Let P (x) be the statement “x + 1 > x. ” What is the truth value of the quantification ∀x. P (x), where the domain consists of all real numbers? Solution: Because P (x) is true for all real numbers x, the quantification ∀x. P(x) is true.

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