lecture4 - Lecture 4 Quantifiers, Nested Quantifiers Rules...

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Lecture 4 Quantifiers, Nested Quantifiers Rules of Inference
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Recap Propositional Function P(x) Universal quantifier: x P(x) “for all values of x, P(x) is true” Existential quantifier: x P(x) “there exists (a value of) x such that P(x) is true” Need to specify the universe/domain! Every variable has to be bound in order to have a valid proposition Negating quatifiers ¬ x P(x) = x ¬P(x) ¬ x P(x) = x ¬P(x)
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Are these equivalent? x (P(x) Q(x)) x P(x) x Q(x) Suppose P(x) is sometimes true, sometimes false, Q(x) is always false
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Suppose the domain is {a,b} with P(a) = T, P(b) = F, Q(a) = Q(b) = F x (P(x) Q(x)) (P(a) Q(a)) (P(b) Q(b)) (T F) (F F) F T F x P(x) x Q(x) (P(a) P(b)) (Q(a) Q(b)) F F T
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Multiple (Nested) quantifiers You can have multiple quantifiers on a statement In the following, domain of x and y are all integers x y P(x, y) “For all x, there exists a y such that P(x,y)” Example: x y (x+y == 0) x y P(x,y) There exists an x such that for all y P(x,y) is true” x y (x*y == 0)
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Order of quantifiers x y and x y are not equivalent! x y P(x,y) P(x,y) = (x+y == 0) is false x y P(x,y) P(x,y) = (x+y == 0) is true
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Negating multiple quantifiers Recall negation rules for single quantifiers: ¬ x P(x) = x ¬P(x) ¬ x P(x) = x ¬P(x) By applying these rules repeatedly, we can negate statements with multiple quantifiers Example: ¬( x y P(x,y)) = x ¬( y P(x,y)) = x y ¬P(x,y)
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Translating between English and quantifiers The product of two negative integers is positive x y ((x<0) (y<0) → (xy > 0))
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lecture4 - Lecture 4 Quantifiers, Nested Quantifiers Rules...

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