Discrete Mathematical Structures by Pearson

Mathematical Logic and Methods of Proofs 4-25

4.8.2 Universal Quantifier

Predicates can be changed into propositions by assigning specific value to their valuables.

Let p(x) be the propositional function defined on the set A. Consider the expression ‘For every

x in A, p(x) is a true statement’.

i.e., (∀ x ∈ A), p(x) is true

or for all x, p(x) is true

Then the symbol ∀, which reads ‘for every’ or ‘for all’, is called the universal quantifier.

In this case T

p

= {x: x ∈ A and p(x) is true} = A

That is, truth set of p(x) is the entire set A.

Example: The function p(x) = x + 2 > 2 over N has universal quantifier as the function is true for

every natural number.

4.8.3 Existential Quantifier

Let p(x) be a propositional function defined on a set A. Consider ...