Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

Propositional logic is pretty limited. Almost anything you really are interested in
requires a more sophisticated form of logic: predicate logic with quantifiers (or
predicate calculus).
Definition: P(x) is a propositional function when a specific value x is substituted
for the expression in P(x) gives us a proposition. The part of the expression
referring to x is known as the predicate.
Examples:
• P(x): x > 24. P(2) = F, P(102) = T.
• P(x): x = y + 1. P(x) = T for one value only (y is an unbounded variable).
• P(x,y): x = y + 1. P(2,1) = T, P(102,-14) = F.
Definition: A statement of the form P(x 1 ,x 2 ,…,x n ) is the value of the
propositional function P at the n-tuple (x 1 ,x 2 ,…,x n ). P is also known as a n-place
(or n-ary) predicate.
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Definition: The universal quantification of P(x) is the statement P(x) is true for
all values of x in some domain, denoted by (x P(x).
Definition: The existential quantification of P(x) is the statement P(x) is true for
at least one value of x in some domain, denoted by )x P(x).
Definition: The uniqueness quantification of P(x) is the statement P(x) is true for
exactly one value of x in some domain, denoted by )!x P(x).
There is an infinite number of quantifiers that can be constructed, but the three
above are among the most important and common.
Examples: Assume x belongs to the real numbers.
• (x 0). The negative real numbers form the domain.
• )!x (x 1223 = 0).
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