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
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…