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Unformatted text preview: 1 Predicate Logic Truth Table p q p q p q p q p q p q p q T T F F T T T T F T F F T F T F F T F T T F F T T F T F F F F F F T T F Precedence of Logical Operators 1. Parentheses 2. negation 3. conjunction 4. disjunction 5. conditionals Quantifiers  The universal quantifier for all False if there exists something that  The existential quantifier one or more or there exits NOTE: ORDER MATTERS FOR QUANTIFIERS NEGATION OF QUANTIFIERS ( x P(x)) x( P(x)) ( x P(x)) x( P(x)) RULES OF INFERENCE RESOLUTION: [(p q) ( p r )] Logically implies q r 2 Proofs TYPES OF PROOFS direct contrapositive contradiction Vacuous Prove the theorm if integer n is odd then n 2 is odd 1. n is odd Premise (h) 2. k (st n=2k+1) Definition of odd(Universe is integers) 3. n=2c+1 for some integer c Step 2, specialization 4. (n=2c+1) (n 2 = (2c+1) 2 Laws of arithemtic 5. n 2 = (2c+1) 2 steps 3 and 4, modus ponens 6. n 2 = 4c 2 + 4c +1 Laws of arithmetic 7. n 2 = 2(2c 2 +2c) +1 Laws of arithmetic 8. k (st n 2 = 2k+1) Step 5, generalization 9. n 2 is odd definition of odd FUN FACTS FOR USE (A(x) B(x)) A(x) B(x) A(x) B(x) (A(x) B(x) 3 Sets Cardinality of a Set Def: Let S be a set. If there are exactly n elements in S where n is an integer we say that S is a finite set and that n is the cardinality of S and we denote it by...
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 Spring '08
 SaktiPramanik

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