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CSS342: Quantifiers

CSS342: Quantifiers. Professor: Munehiro Fukuda. 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”:

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CSS342: Quantifiers

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  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 A • 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 CSS342: Quantifiers

  5. Universally Quantified StatementsExample 1 • For every real number x, if x > 1, then x + 1 > 1 is true. • To be true, we need to consider all cases: x ≤ 1 and x > 1 • Proof: (1 ) If x ≤ 1, the hypothesis x > 1 is false, thus the conditional proposition is true. (2) If x > 1, x + 1 > x (always true) since x > 1, x + 1 > x > 1 Thus, the conditional proposition is true. Therefore, this universally quantified statement is true. CSS342: Quantifiers

  6. Universally Quantified StatementsExample 2 • For every real number x, x2 – 1 > 0 is false. • To be false, we only need to show a counterexample: x = 1 • Proof: If x = 1, the proposition 12 – 1 > 0 is false The value 1 is a counterexample. Thus, this universally quantified statement is false CSS342: Quantifiers

  7. Existentially Quantified Statements E • x, P(x) • Meaning: for some x, P(x), for at least one x, P(x), or there exists x such that, P(x) • : an existential quantifier • P(x): a existentially qualified statement • X: a bound variable, (i.e., bound by the quantifier ) • is true: if P(x) is true for at least one x in D • is false: if P(x) is false for every x in D E E CSS342: Quantifiers

  8. Existentially Quantified StatementsExample 1 • For some positive integer n, if n is prime, then n + 1, n + 2, n + 3, and n + 4 are not prime. is true. • To be true, we only need to show at least one case makes it true: x = 1 • Proof: If n = 23, n + 1 = 24 (=3 * 8) is not prime n + 2 = 25 (=5 * 5) is not prime n + 3 = 26 (=2 * 13) is not prime n + 4 = 27 (=3 * 9) is not prime Thus, the proposition is true. Therefor, this existentially quantified statement is true. CSS342: Quantifiers

  9. Existentially Quantified StatementsExample 2 • For some real number x, 1 / (x2 + 1) > 1 is false. • To be false, we need to consider all cases: This in turn means that for all real number x, 1 / (x2 + 1) ≤ 1 • Proof: Since 0 ≤ x2 for every real number x, 1 ≤ x2 + 1 By dividing both sides of this inequality expression by x2 + 1 1 / (x2 + 1) ≤ 1 Is true. Therefore, 1 / (x2 + 1) >1 is false for every real number x. CSS342: Quantifiers

  10. Domain D Domain D P(x) ≠ true(false) P(x) = true P(x) = true ≡ P(x) = true P(x) = true P(x) = true A E xP(x) and xP(x) • If it is not the case that P(x) is true for every x, there is a counterexample showing that P(x) is not true for at least one x. CSS342: Quantifiers

  11. E A xP(x) and xP(x) • If it is not the case that P(x) is true for some x, P(x) is false for every x. Domain D Domain D P(x) = true P(x) = false P(x) = false ≡ P(x) = false P(x) = false P(x) = false CSS342: Quantifiers

  12. Generalized De Morgan Laws for Logic If P is a propositional funciton, each pair of propositions in (a) and (b) is logically equivalent, (i.e., has the same truth values.) • xP(x) and xP(x) • xP(x) and xP(x) A E E A CSS342: Quantifiers

  13. Generalized De Morgan Laws for Logic (Cont’d) • For every x, P(x) means • P(1) && P(2) && … && P(n) • For some x, P(x) means • P(1) || P(2) || … || P(n) • x, P(x) ≡ x, P(x) means • !(P(1) && P(2) && … && P(n)) ≡ !P(1) || !P(2) || … || !P(n) • x, P(x) ≡ x, P(x) means • !(P(1) || P(2) || … || P(n)) ≡ !P(1) && !P(2) && … && !P(n)) A E E A CSS342: Quantifiers

  14. Revisiting Existentially Quantified Statement Example 2 • For some real number x, 1 / (x2 + 1) > 1 is false. • Let P(x) be 1 / (x2 + 1) > 1, then x, P(x) • According to De Morgan Laws, we may prove x, P(x) This in turn means that for all real number x, 1 / (x2 + 1) > 1 is false, (i.e., 1 / (x2 + 1) ≤ 1) • Proof: E A Since 0 ≤ x2 for every real number x, 1 ≤ x2 + 1 By dividing both sides of this inequality expression by x2 + 1 1 / (x2 + 1) ≤ 1 CSS342: Quantifiers

  15. Two-Variable Propositional Function • P(x, y): x + y = 0 • x, y, x + y = 0 is true. • Proof: for any x, we can find at least y = -x. Thus, x + y = x – x = 0 • y, x, x + y = 0 is false. • Proof: if it is true, the statement can be replaced with constant Y x, x + Y = 0 However, we can choose x = 1 – Y, such that x + Y = 1 – Y + Y = 1 ≠ 0. A E E A A CSS342: Quantifiers

  16. The Logic Game • Given a propositional function, P(x, y) You and your opponent play a logic game. • Your goal: make P(x, y) true • You can: choose a bound variable of to make it true • Your opponent, Farley’s goal: make P(x, y) false • Farley can: choose a bound variable of to make it false • Play with P(x, y): x + y = 0 E A CSS342: Quantifiers

  17. The Logic Game (Cont’d) A E • x, y, P(x, y): • No matter what Farley chooses for x, you choose y = -x. • You win. P(x, y) is true. • x, y, P(x, y): • Regardless of what you choose for x, Farley chooses y = 1- x. • Farley wins. P(x, y) is false. • x, y, P(x, y): • Farley chooses x and y such that x + y ≠ 0, (e.g., x = 0, y = 1) • Farley wins. P(x, y) is false. • x, y, P(x, y) • You choose x and y such that x + y = 0, (e.g., y = -x or x=1, y= -1) • You win. P(x, y) is true. E A A A E E CSS342: Quantifiers

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