This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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...
View
Full
Document
This note was uploaded on 01/29/2012 for the course CSE 260 taught by Professor Saktipramanik during the Spring '08 term at Michigan State University.
 Spring '08
 SaktiPramanik

Click to edit the document details