7 - Quantifiers Let P ( x ) be a statement depending on x ....

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Unformatted text preview: Quantifiers Let P ( x ) be a statement depending on x . Then x , P ( x ) is For all x , P ( x ) is TRUE. x , P ( x ) is There exists an x such that P ( x ) is TRUE. Example. Let P ( x ) be x 2 = 2. Is x , P ( x ) TRUE or FALSE? Is x , P ( x ) TRUE or FALSE? 1 Example. Let P ( x ) be x 2 = 2. A = x , P ( x ). B = x , P ( x ). We need to specify the universe of discourse for x . 1. U of D = Q . A and B are FALSE. 2. U of D = { 2 ,- 2 } . A and B are TRUE. 3. U of D = R . A is FALSE, B is TRUE. 2 If U of D is P then x,y x is equivalent to y = 1. ( y is a least element in P ). y x,y x is TRUE. (There is a least element in P ). y x,y x is FALSE. (Any positive integer is least in P ). 3 Let S be some U of D. Then x S , P ( x ) means For all x from S , P ( x ) is TRUE x S , P ( x ) means There is x from S such that P ( x...
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This note was uploaded on 02/05/2010 for the course MATH 135 taught by Professor Andrewchilds during the Spring '08 term at Waterloo.

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7 - Quantifiers Let P ( x ) be a statement depending on x ....

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