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06+Slides--Rules+of+Inference

# 06+Slides--Rules+of+Inference - CS103 HO#6 Slides-Rules of...

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CS103 HO#6 Slides--Rules of Inference April 1, 2011 1 CS103 Mathematical Foundations of Computing 4/1/11 Homework can be submitted in the filing cabinet in the lobby near my office. There is a drawer marked CS103. Universe of Discourse HasTaken(a, b) student a has taken class b If we write x y HasTaken (x, y) , it is understood that the quantifiers operate over the appropriate domains. 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) L(x, y) : x loves y Everybody loves Fred. Everybody loves somebody. There is somebody whom nobody loves.

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CS103 HO#6 Slides--Rules of Inference April 1, 2011 2 L(x, y) : x loves y Everybody loves Fred. x L(x, Fred) Everybody loves somebody. x y L(x, y) There is somebody whom nobody loves. x y ¬L(y, x) x ¬ y L(y, x) ¬ x y L(y, x) Every cube is to the left of every tetrahedron. x (Cube(x)   y (Tet(y) LeftOf(x, y))) Every cube is to the left of every tetrahedron. x (Cube(x)   y (Tet(y) LeftOf(x, y))) x y (Cube(x) (Tet(y) LeftOf(x, y))) x y ((Cube(x) (Tet(y)) LeftOf(x, y))) Every small cube is in back of a large cube. Some cube is in front of every tetrahedron. Anything with something in back of it is a cube. x ( y BackOf(y, x) Cube(x) ) x ( ¬ y BackOf(y, x) Cube(x) ) x ( y ¬ BackOf(y, x) Cube(x) ) x y ( ¬ BackOf(y, x) Cube(x) ) x P(x)   y Q(y) ¬ x P(x)   y Q(y) x ¬P(x)
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06+Slides--Rules+of+Inference - CS103 HO#6 Slides-Rules of...

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