Final Review Sheet - 1 Predicate Logic Truth Table p q p...

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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)
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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 | S |
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