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Lec05_FirstOrderLogic__part_2_ - TDS1191 Discrete...

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[Lecture05] [First Order Logic (part 2)] Discrete Structures 1 TDS1191 Discrete Structures Lecture 05 First Order Logic (part 2) Multimedia University Trimester 2 Session 2011/2012
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[Lecture05] [First Order Logic (part 2)] Discrete Structures 2 The Order of Quantifiers Statement When True ? When False? 2200 x 2200 y P(x,y) 2200 y 2200 x P(x,y) P(x,y) is true for every pair of x, y . There is a pair x, y for which P(x,y) is false 2200 x 5 y P(x,y) For every x there is a y for which P(x,y) is true There is an x such that P(x,y) is false for every y 5 x 2200 y P(x,y) There is an x for which P(x,y) is true for every y For every x there is a y for which P(x,y) is false 5 x 5 y P(x,y) 5 y 5 x P(x,y) There is pair x, y for which P(x,y) is true P(x,y) is false for every pair of x, y
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[Lecture05] [First Order Logic (part 2)] Discrete Structures 3 Translating Mathematical Statements into Logical Expressions Involving Nested Quantifiers Translate the statement “the sum of two positive integers is always positive” into logical expression. 2200 x 2200 y (( x > 0) ( y > 0) ( x + y > 0)) Translate the statement “the product of a positive real number and a negative real number is a negative real number” into logical expression. 2200 x 2200 y (( x > 0) ( y < 0) ( xy < 0)) Translate the statement “every nonzero real number has a reciprocal” into logical expression. 2200 x (( x ≠ 0) ( 5 y ( y = 1/ x )))
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[Lecture05] [First Order Logic (part 2)] Discrete Structures 4 Translate the following statements into English. a) 2200 x 5 y F(x,y) b) 5 x 2200 y F(x,y) c) 5 x 2200 y 2200 z (F(x,y) F(x,z) ( y z ) ¬ F(y,z)) d) 2200 x 5 y 5 z (F(x,y) F(x,z) ( y z ) ¬ F(y,z)) Here the domain of discourse is all the students in MMU, and F(a,b) means a and b are friends. Translate the following statements into English. a) 2200 x 5 y F(x,y) b) 5 x 2200 y F(x,y) c) 5 x 2200 y 2200 z (F(x,y) F(x,z) ( y z ) ¬ F(y,z)) d) 2200 x 5 y 5 z (F(x,y) F(x,z) ( y z ) ¬ F(y,z)) Here the domain of discourse is all the students in MMU, and F(a,b) means a and b are friends. Translating Quantifiers into English Solution a) Every student in MMU has some friends. b) There is a student in MMU who is a friend of all the other students in MMU. c) There is a student in MMU where none of his friends are friends with each other. d) Every student in MMU has at least two friends that are not friends of each other. a) Every student in MMU has some friends. b) There is a student in MMU who is a friend of all the other students in MMU. c) There is a student in MMU where none of his friends are friends with each other. d) Every student in MMU has at least two friends that are not friends of each other.
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[Lecture05] [First Order Logic (part 2)] Discrete Structures 5 Translate the following two statements into English a) 2200 x 5 y ( x + y = 0) b) 5 y 2200 x ( x + y = 0) Do they express the same thing? Here the domain of discourse is the set of real numbers.
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