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handout2 - ECS20 Handout 2 January 6 2011 Propositional...

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ECS20 Handout 2 January 6, 2011 Propositional function – predicate and quantifier 1. Propositional function P ( x ): a statement involving the variables x Examples Let P ( x ) denote “ x > 3”, what are the truth values of P (4) and P (2)? Let Q ( x, y ) denote “ x = y + 3”, what are the truth values of the proposition Q (1 , 2) and Q (3 , 0)? 2. A statement of the form P ( x 1 , x 2 , . . . , x n ) is the value of the propositional function P at the n -tuple ( x 1 , x 2 , . . . , x n ), and P is called the predicate . 3. The domain of discourse , denoted as U , is the set of values x that x is allowed to take in P ( x ). 4. The universal quantification of P ( x ) is the proposition P ( x ) is true for all value of x in U ”. Notation: x P ( x ), is called the universal quantifier. Truth values of xP ( x ) xP ( x )” = True, when P ( x ) is true for every x in U . xP ( x )” = False, when there is an x in U for which P ( x ) is false. 5. Examples: “for all integers n , 2 n is even” ( True ) “for all real numbers x , x 2 - 1 > 0” ( False, e.g. x = 0 )
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