Discrete Mathematics

CS 2610

August 21, 2008

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Agenda

Nested quantifiers

Rules of inference

Proofs

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Review

A predicate

P

, or propositional function, is a

function that maps objects in the universe of

discourse to propositions

Predicates can be quantified using the universal

quantifier (“for all”) ∀ or the existential quantifier

(“there exists”) ∃

Quantified predicates can be negated as follows

¬∀x P(x) ≡ ∃x ¬P(x)

¬∃x P(x) ≡ ∀x ¬P(x)

Quantified variables are called “bound”

Variables that are not quantified are called “free”

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Predicate Logic and Propositions

An expression with zero free variables is an

actual proposition

Ex. Q(x) : x > 0, R(y): y < 10

∃ x Q(x) ∧ ∃y R(y)

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