Math

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Math

  1. 1. TAF 3023: DISCRETEMATH Power !
  2. 2. PROPOSITIONAL LOGICProposition • possible condition of the world about which we want to say something.Example • Apple is a fruit.
  3. 3. PROPOSITIONAL LOGIC• Propositional • A proposition or statement is a logic sentence which is either true or false.• Example • True - Apple is a fruit. • False - Rice is a fruit.
  4. 4. PROPOSITIONAL LOGIC• Propositional • Propositional variables use letters to variables represent it, just as letters used to represent numerical variables.• Example • p, q, r, s, ………
  5. 5. PROPOSITIONAL LOGICTypes of Truth Table • Negation • Conjunction Proposition Proposition
  6. 6. PROPOSITIONAL LOGICTypes of Truth Table • Disjunction • Exclusive Or of Two Proposition Propositions
  7. 7. PROPOSITIONAL LOGICTypes of Truth Table• Conditional Statement and Biconditional Statement•
  8. 8. PROPOSITIONAL LOGIC Types of Truth Table• Compound Proposition•
  9. 9. PROPOSITIONAL EQUIVALENCE• Tautology  a compound proposition that is always true.• Contradiction  a compound proposition that is always false.• Contingency  a compound proposition that contain neither true or false that mean in its truth table have at least one true and at least one false.
  10. 10. PROPOSITIONAL EQUIVALENCE Examples: p ~p p V ~p • Tautology T F T F T T • Contradiction p ~p p ^ ~p T F F F T F • Contingency p ~p p  ~p T F T F T F
  11. 11. LOGICAL EQUIVALENCE• In logic, statements p and q are logically equivalent if they have the same logical content• (Mendelson 1979:56) two statements are equivalent if they have the same truth value in every model• Logical Equivalence Table
  12. 12. LOGICAL EQUIVALENCEDe Morgan’s Law• Probably the most important logical equivalence  ¬(p ∧ q) ≡ ¬p ∨¬q  ¬(p ∨ q) ≡ ¬p ∧¬q
  13. 13. PREDICATE AND QUANTIFIERS Introduction:• Predicate is an open statement or sentence that contains a finite numbers of variables. Predicates become statement when specifies values are substituted for the variables by certain allowable choices of value. • variable x - subject Example: • greater than 3 – predicate “x is greater than 3” • predicate in the form of:  P(x) – this is a unary predicate (has one OR variable)  P( x, y) – this is a binary predicate (has • denote as P(x) two variables)  P(x1, x2, x…….., xn) – this is an n-ary or n- place predicate – (has n individual variables in a predicate)
  14. 14. PREDICATE AND QUANTIFIERS Quantifiers:• Definition • a logical symbol which makes an assertion about the set of values which make one or more formulas true. • universal quantifier: read for “all”, “each”, “every”. • existential quantifier: read for “some” statement that is true or false.• Example • universal - “Everyone likes cakes“. “Not everyone likes cakes”. • existential - “Someone likes cakes”. “No one likes cakes”.
  15. 15. PREDICATE AND QUANTIFIERS Examples Using Quantifiers: Universal and Existential Quantifier Statement: True: False: ∀xP(x) P(x) is true for every There is an x for x. which P(x) is false. ∃xP(x) There is an x for P(x) is false for every which P(x) is true. x.
  16. 16. PREDICATE AND QUANTIFIERS Examples Using Quantifiers: Universal and Existential Quantifier Statement: True: False: ∀xP(x) x+1>x x<2 If P(x) = 1, the If P(x) = 1 or 0, the quantification is quantification is true. true. But If P(x) = 3, the quantification is false. ∃xP(x) x>3 x=x+1 If P(x) = 4, the P(x) is false for all real quantification is number. true.

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