Propositional Function: P(x)

A sentence expressed in a way that would assume the value of true or false, except that within the sentence there is a variable (x) that is not defined or specified, which leaves the statement undetermined. (According to Wikipedia)

Domain of Discourse: D

Scope of values for the variable included in a propositional function

Examples

P(x): x^2 -3x = 0 Determine the truth values for P(1) and P(3)

P(1): -2 ≠ 0 False

P(3): 0 = 0 True

Q(x,y): x = 2y Determine the truth values for Q(1,2) and Q(2,1)

Q(1,2): 1 ≠ 4 False Q(2,1): 2 =2 True

Quantifiers

Universal Quantifier: ∀

All values included in the Domain of Disclose(D) for propositional function. P(x): ∀x P(x)

Examples

When Domain of Disclose(D) is Integers(Z), find the truth values for the propositional functions.

P(x): x is a real number

∀x P(x): for every integer x, x is a real number. True

P(x): x is a natural number

∀x P(x): for every integer x, x is a natural number. False

P(x): x is an imaginary number

∀x P(x): for every integer x, x is an imaginary number. False

Existential Quantifier: ∃

At least one value included in the Domain of Disclose(D) for propositional function. P(x): ∃x P(x)

Examples

When Domain of Disclose(D) is Integers(Z), find the truth values for the propositional functions.

P(x): x is a real number

∃x P(x): there exists an integer x such that x is a real number. True

P(x): x is a natural number

∃x P(x): there exists an integer x such that x is a natural. True

P(x): x is an imaginary number

∃x P(x): there exists an integer x such that x is an imaginary number. False

Quantifiers with logic operations