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07 Slides--Rules of Inference

# 07 Slides--Rules of Inference - CS103 HO#7 Slides-Rules of...

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CS103 HO#7 Slides--Rules of Inference April 5, 2010 1 CS103 Mathematical Foundations of Computing 4/5/10 Homework can be submitted in the filing cabinet in the lobby near my office. There is a drawer marked CS103. Notice that implication in a sentence like x (P(x) Q(x)) seems quite natural. But what does x (P(x) Q(x)) mean? It's better to write the second sentence as x (¬P(x) Q(x)) Translations S(x): x is a student in this class C(x): x has visited Canada M(x): x has visited Mexico Some student in this class has visited Mexico. x ( S(x) M(x) ) or is it x ( S(x) M(x) ) Every student in the class has visited Canada or Mexico. x (S(x) [ C(x) M(x) ] ) Multiple Quantifiers x y (Boy(x) Girl(y) Likes(x, y)) x y [(Tet(x) Dodec(y)) Smaller(x, y)] x y ((Cube(x) Cube(y)) . . .) Could x and y be the same cube? Multiple Quantifiers x y (Boy(x) Girl(y) Likes(x, y)) x y [(Tet(x) Dodec(y)) Smaller(x, y)] x y ((Cube(x) Cube(y) x ≠ y) . . .) x y (Cube(x) Cube(y) x ≠ y) Universe of Discourse HasTaken(a, b) a has taken class b If we write x y HasTaken (x, y) , it is understood that the quantifiers operate over the appropriate domains.

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CS103 HO#7 Slides--Rules of Inference April 5, 2010 2 x y HasTaken (x, y) x y HasTaken (x, y) x y HasTaken (x, y) x y HasTaken (x, y) y x HasTaken (x, y) y x HasTaken (x, y) Q(x,y) : x + y = 0 y x Q(x, y) x y Q(x, y) Summary of Quantifiers in Arity-Two Predicates Statement When true? When false? x y P(x,y) P(x,y) is true for all pairs (x,y) There is a pair (x,y) for which P(x,y) is false x y P(x,y) For every x, there is a y for which There is an x such that P(x,y) P(x,y) is true is false for every y x y P(x,y) There is an x for which P(x,y) is For every x, there is a y for true for every y which P(x,y) is false x y P(x,y) There is a pair (x,y) for which P(x,y) is false for all pairs (x,y) P(x,y) is true y x P(x,y) y x P(x,y) What if we wanted to test alternatives to determine if x y P(x, y) is true?
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