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05+Slides--Quantifiers

# 05+Slides--Quantifiers - CS103 HO#5 Quantifiers...

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CS103 HO#5 Quantifiers 3/30/11 1 CS103 Mathematical Foundations of Computing 3/30/11 Policy for hard copy handouts: Lecture slides will be provided in hard copy form; most other handouts (including assignments) will be on CourseWork only. Please pick up a copy of the slides only if you plan to use it. We will adjust the number printed accordingly. Conditionals p q p q ¬ p q T T T T T F F F F T T T F F T T The best rendering of p q in English is " if p , then q " p q p q ¬ p q T T T T T F F F F T T T F F T T If the moon is not made of cheese, then triangles have three sides. If the moon is made of cheese, then triangles have three sides. If the moon is made of cheese, then triangles are round. If the moon is made of cheese, then the moon is not made of cheese. It will snow only if it is cold. Cold is a necessary condition for snow. Snow Cold If it's not cold, it will not snow. ¬Cold ¬Snow p only if q is translated as p q If p is true, then q must be true, because that is the only way p can be true. Contrapositive Biconditional : true when p and q have the same truth value p q p q ( q p ) ( p q ) T T T T T F F F F T F F F F T T iff p only if q is translated as p q p if q is translated as q p p if and only if q is translated as ( q p ) ( p q ) p q is equivalent to ( q p ) ( p q ) p q ( q p ) ( p q ) Biconditional

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CS103 HO#5 Quantifiers 3/30/11 2 p q r ................................. T T T T T T F T T F T T T F F T F T T T F T F T F F T T F F F T Main connective What if everything under the main connective is a T? p q r ................................. T T T T T T F T T F T T T F F T F T T T F T F T F F T T F F F T Tautology Main connective A tautology is a compound proposition that cannot be false , due to its structure and the meanings of the truth-functional connectives it contains. This tautology is know as the " Law of the Excluded Middle ". It says that every proposition is either true or false. In first order logic, there is no "maybe". Tautology p p   p T T F F T T Contradiction p p   p T F F F F T A contradiction is a compound proposition that cannot be true , due to its structure and the meanings of the truth-functional connectives it contains. When would p q be false? Answer 1: when it is not true Answer 2: when both p and q are false. So ( p q) p   q Similarly ( p q) p   q De Morgan's Laws Idempotent Laws p p p p p p Double Negation ( p ) p Commutative Laws p q q p p q q p Associative Laws ( p q ) r p ( q r ) ( p q ) r p ( q r ) Distributive Laws p ( q r) ( p q ) ( p r ) p ( q r ) ( p q )
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05+Slides--Quantifiers - CS103 HO#5 Quantifiers...

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