Quantifiers

Quantifiers - CSS342: Quantifiers 1 CSS342: Quantifiers...

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Unformatted text preview: CSS342: Quantifiers 1 CSS342: Quantifiers Professor: Munehiro Fukuda CSS342: Quantifiers 2 Review of Propositions • Proposition: a statement that is either true or false, but not both • Example: – 1 < 4 is true. – 2 > 5 is false. – 3 is an odd number – Then, how about x is an odd number ? • The statement “X is an odd number”: – is true if x = 103 – is false if x = 8 – Most of the statements in math and CS use variables. • We need to extend the system of logic! CSS342: Quantifiers 3 Propositional Functions • P(x): a statement involving the variable x – Example: x is an odd number. – P(x) itself is not a proposition. – For each x in the domain D of discourse of P, if D is the set of positive integers, P(x) is a proposition • P(1): 1 is an odd number (= true). • P(2): 2 is an odd number (= false). • … • Either true or false • X: a free variable, (i.e., free to roam over the domain D) • How about for every x or for some x, P(x) is true/false? – Most statements in math and CS use such phrases. CSS342: Quantifiers 4 Universally Quantified Statements • x, P(x) – Meaning: for every x, P(x), for all x, P(x), or for any x, P(x) – : a universal quantifier – P(x): a universally qualified statement – X: a bound variable, (i.e., bound by the quantifier ) – is true: if P(x) is true for every x in D – is false: if P(x) is false for at least one x in D A A A CSS342: Quantifiers 5 Universally Quantified Statements Example 1 • For every real number x, if x > 1, then x + 1 > 1 is true....
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Quantifiers - CSS342: Quantifiers 1 CSS342: Quantifiers...

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