Math 51/COEN 19 Day 3, 1.4 Quantifiers 1. 3 Predicates A lot like functions that return booleans Let...
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Transcript of Math 51/COEN 19 Day 3, 1.4 Quantifiers 1. 3 Predicates A lot like functions that return booleans Let...
Math 51/COEN 19 Day 3, 1.4 Quantifiers
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3
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) =
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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
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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
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Negation
Precedence
Operator Precedence
¬ 1
2
3
→ 4
5
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Quantifiers beat them all. Lesson: Use parentheses
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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
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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
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Uniqueness quantifier
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Logical equivalence with quantifiers