Math 51/COEN 19 Day 3, 1.4 Quantifiers 1

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Predicates A lot like functions that return booleans Let P(x) denote x<12 – P(2) = – P(50) = Let P(x, y, z) denote x-y<z – P(5, 4, 2) = – P(10, 5, 1) = 4

Quantifiers Allow us to reason about the way predicates behave over a domain (don’t forget the domain!) “Universal quantifier” – For all, for every, all of, for each, given any, for arbitrary, for each, for any “Existential quantifier” – There exists, for some, for at least one, there is P(x) denotes x-x=0 and Q(x) denotes x+2=8 where x is an integer 5

Relationship to propositional logic P(x) denotes x<10, domain positive integers less than 5 Universal quantifier is like a big conjunction Existential quantifier is like 6

Negation 7

Precedence OperatorPrecedence ¬1 2 3 →4 5 8 Quantifiers beat them all. Lesson: Use parentheses

Binding of variables and scope Variables tied to a quantifier are bound, while those not tied to a quantifier are free. The part of the logical expression the quantifier applies to is the scope. We often reuse variable names, so pay attention to scope 9

English to logic with quantifiers! There’s a girl with a crown and a sword No one knows if P=NP For every integer x, if x is an odd square then x is one more than a multiple of 4 The lights are off when everyone is out of the room 10

Uniqueness quantifier 11

Logical equivalence with quantifiers 12