Quantifier
- 2. What is Quantification?? =>> Quantification is a method to transform a propositional function into a proposition. Express the extent to which a predicate is true.
- 3. Types 1. Universal quantification 2. Existential quantification
- 4. Universal quantification?? A predicate is true for every element under consideration denoted as ∀ p(x) Universal Quantification allows us to capture statements of the form “for all” or “for every”.
- 5. Pros and cons of Universal quantification • “p(x) for all values of x in the domain” • Read it as “for all x p(x)” or “for every x p(x)” • A statement is false if and only if p(x) is not always true. • An element for which p(x) is false is called a counterexample of ∀ p(x) • A single counterexample is all we need to establish that is not true .
- 6. Existential quantification?? There exists an element x in the universe of discourse. denoted as ∃x P (x )
- 7. Pros & Cons of Existential quantification • “There exists an element x in the domain such that p(x) (is true)” • Denote that as ∃x P (x ) where ∃ is the existential quantifier. • In English, “for some”, “for at least one”, or “there is”. • Read as “There is an x such that p(x)”, “There is at least one x such that p(x)”, or “For some x, p(x)”.
- 8. Example Universal :: ∀x (X2 ≥ 0) --> "the square of any number is not negative.'' Existential :: ∃x (X ≥ X2 ) --> is true since x=0 is a solution and there are many others.
- 9. Precedence of quantifiers ∀ and ∃ have higher precedence than all logical operators from propositional calculus For example ∀x P (x ) ∨ Q (x ) means (∀x P (x )) ∨ Q (x ), not ∀x (P (x ) ∨ Q (x )).
- 10. Thank You