For every x px true if px is true for every x false

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” ; “for every x P(x) true if P(x) is true for every x false if P(x) is false for at least one x in the ) ( x xP 2200 2200 ) ( x xP 2200
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LOGICAL STRUCTURE Examples: Domain for x is the set of real numbers 1. P(x): “x+1>x” 2. P(x): “x>5” 3. Q(x): “x 2 >= x” Note: is false if P(x) is false for at least one x in D. A value x in the domain of discourse that makes P(x) false is called a counterexample to the statement . ) ( x xP 2200 ) ( x xP 2200
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LOGICAL STRUCTURE Definition The existential quantification of P(x) is the proposition “There exist an element x in the domain of discourse such that P(x) is true. Notation: denotes the existential quantification of x is called the existential quantifier read as “there is an x s.t. P(x)” “there is at least one x s.t. P(x)” ) ( x xP 5 5 ) ( x xP 5
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LOGICAL STRUCTURE true if P(x) is true for at least one x in the domain false if P(x) is false for every x in the domain Examples: Domain is the set of real numbers 1. P(x): x>3 2. Q(x): x = x+1 3. P(x): 1 2 + x x
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LOGICAL STRUCTURE Exercises. Give the domain. 1. 2. True or False. Domain is the set of integers. P(x): x>2 Q(x): x<2 1. 2. ) 1 ( 2 = 2200 x x
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LOGICAL STRUCTURE Negation with Quantifiers example : Every student in the class has taken a course in calculus. P(x) :“x has taken a course in calculus” Negation of : It is not the case that every student in the ) ( ) ( x P x x xP 5 2200 ) ( x xP 2200 ) ( x xP 2200 ) ( x xP 2200 ) ( x P x 5
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LOGICAL STRUCTURE Example :There is a student in this class who has taken a course in calculus. Q(x) :“x has taken a course in calculus” Negation of It is not the case that there is a student in this class who has taken a course in ) ( ) ( x Q x x xQ 2200 5 ) ( x xQ 5 ) ( : ) ( x xQ x xQ 5 5 ) ( x Q x 2200
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LOGICAL STRUCTURE RULES of INFERENCE Definition (Argument) An argument is a sequence of propositions written as. The propositions p 1 ,p 2 ,…,p n are called the hypotheses (or premises), and the proposition q is called the conclusion. The argument is valid provided that if p 1 andp 2 and p n are all true, then q must also be true; otherwise, the argument is invalid (or a fallacy).
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LOGICAL STRUCTURE Rules of inference- brief and valid argument used within a larger argument such as proof Example Determine whether the argument is valid. p→q p q D:\DiscreteMath\QUANTIFIERS.doc
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LOGICAL STRUCTURE Example (A Logical Argument) If I dance all night, then I get tired. I danced all night. Therefore I got tired. Logical representation of the underlying variables: p: I dance all night. q: I get tired. Logical analysis of the argument: p→q p
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LOGICAL STRUCTURE If I dance all night, then I get tired. I got tired. Therefore I danced all night. Logical form of argument: p→q q p
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LOGICAL STRUCTURE Examples: 1. Show that the hypotheses “ If you send me an email message, then I will finish writing the program,” If you do not send me an email message, then I will go to sleep early,” and “If I go to sleep early, then I will wake up feeling refreshed “ lead to the conclusion “If I do not finish writing the program, then I will wake up feeling refreshed.”
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