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Slide subtopic 3 Slide subtopic 3 Presentation Transcript

  • UNIVERSITI PENDIDKAN SULTAN IDRIS SUBTOPIC 3 QUANTIFIERS PREPARED BY : MOHAMAD AL FAIZ BIN SELAMAT
  • IntroductionA proposition is a statement; either “true” or “false”.The statement P : “n” is odd integer.The statement P is not proposition because whether p istrue or false depends on the value of n
  • Topic1 • Quantifiers2 • Universal Quantification • Counterexample3 • Existential Quantification45 • De Morgan’s Law For Logic
  • 1. QuantifiersDefinition: Let P (x) be a statement involving the variable x and let D be a set. We called P a proportional function or predicate (with respect to D ) , if for each x ∈ D , P (x) is a proposition. We called D the domain of discourse of P.
  • Example 1Let P(n) be the statement n is an odd integerFor example: If n = 1, we obtain the proposition. P (1): 1 is an odd integer (Which is true) If n = 2, we obtain the proposition P (2): 2 is an odd integer (Which is false)
  • 2. Universal Quantification Definition: Let P be a propositional function with the domain of discourse D. The universal quantification of P (x) is the statement. “For all values of x, P is true.” ∀x, P (x) Similar expressions: For each… For every… For any…
  • 3. CounterexampleDefinition : A counterexample is an example chosen to show that a universal statement is FALSE. To verify : ∀x, P (x) is true ∀x, P (x) is false
  • Example 2
  • 4. Existential Quantification Let P be a proportional function with the domain of discourse D. The existential quantification of P (x) is the statement. “there exist a value of x for which P (x) is true. ∃x, P(x) Similar expressions : - There is some… - There exist…
  • Example 3
  • 5. De Morgan’s Law For Logic Theorem: (∀x, P (x)) ≡ (∃x, (P(x)) (∃x, (P(x)) ≡ (∀x, P (x)) The statement “The sum of any two positive real numbers is positive”. ∀x > 0∀y > 0
  • Example 4