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Quantifiers

Quantifiers. Supplementary Notes. Prepared by Raymond Wong. Presented by Raymond Wong. e.g.1 (Page 6). We are going to prove the following claim C is true : statement P(m) is true for each non-negative integer m, namely 0, 1, 2, …. P(0). true.

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Quantifiers

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  1. Quantifiers Supplementary Notes Prepared by Raymond Wong Presented by Raymond Wong

  2. e.g.1 (Page 6) We are going to prove the following claim C is true: statement P(m) is true for each non-negative integer m, namely 0, 1, 2, … P(0) true If we can prove that statement P(m) is true for each non-negative integer separately, then we can prove the above claim C is correct. P(1) true P(2) true P(3) true P(4) true … true

  3. e.g.1 We are going to prove the following claim C is false: statement P(m) is true for each non-negative integer m, namely 0, 1, 2, … P(0) false true P(1) true There may exist another non-negative integer k such that P(k) is false P(2) true P(3) true P(4) true … true

  4. P(3) true P(4) true … true e.g.1 We are going to prove the following claim C is false: statement P(m) is true for each non-negative integer m, namely 0, 1, 2, … 0, 1, 2, … -1, -2,… m2 > m integer 02 > 0 P(0) false 12 > 1 P(1) false 22 > 2 P(2) true

  5. e.g.2 (Page 9) We are going to prove the following claim C is true: there exists a non-negative integer m such that statement P(m) is true P(0) P(1) If we can prove that statement P(m) is true for ONE non-negative integer, then we can prove the above claim C is correct. P(2) true P(3) P(4) …

  6. e.g.2 We are going to prove the following claim C is false: there exists a non-negative integer m such that statement P(m) is true P(0) false P(1) false If we can prove that statement P(m) is false for each non-negative integer separately, then we can prove the above claim C is false. P(2) false P(3) false P(4) false … false

  7. P(0) P(1) P(2) P(3) P(4) … e.g.2 We are going to prove the following claim C is true: there exists a non-negative integer m such that statement P(m) is true m2 > m integer 22 > 2 true

  8. e.g.3 (Page 13) • E.g. Using the quantifier notations, please re-write the Euclid’s division theorem that states For every positive integer n and every non-negative integer m, there are integers q and r, with 0  r < n such that m = qn + r.

  9. For every positive integer n and every non-negative integer m, there are integers q and r, with 0  r < n such that m = qn + r. e.g.3 For every positive integer n and every non-negative integer m, there are non-negative integers q and r, with r < n such that m = qn + r. Since m is non-negativeand n is a positive integer, we derive that q and r are also non-negative. Let Z+ be the set of positive integers. Let N be the set of non-negative integers. m  N ( ) q  N ( ) r  N ( ) (r < n)  (m = qn + r) n  Z+ ( )

  10. e.g.4 (Page 15) m  N ( ) q  N ( ) r  N ( ) (r < n)  (m = qn + r) n  Z+ ( ) Let p(m, n, q, r) denote m = nq + r with r < n If we remove the universe, then we can see the order in which the quantifieroccurs m ( ) q ( ) r p(m, n, q, r) n ( )

  11. e.g.5 (Page 19) • Is the following statement true? • x  R+ (x > 1) If this statement is correct, we need to prove the following. Let P(x) be “x > 1” P(0) true P(0.1) true P(0.2) true … true P(1) true … true

  12. P(0) P(0.1) P(0.2) … P(1) … e.g.5 • Is the following statement true? • x  R+ (x > 1) If this statement is incorrect, we need to prove the following. Let P(x) be “x > 1” false

  13. If this statement is incorrect, we need to prove the following. Let P(x) be “x > 1” P(0) P(0.1) false P(0.2) … P(1) … e.g.5 • Is the following statement true? • x  R+ (x > 1) Consider x = 0.1 Note that 0.1  R+ “0.1 > 1” is false. This statement is false.

  14. e.g.6 (Page 19) • Is the following statement true? • x  R+ (x > 1) If this statement is correct, we need to prove the following. Let P(x) be “x > 1” P(0) P(0.1) P(0.2) true … P(2) …

  15. e.g.6 • Is the following statement true? • x  R+ (x > 1) If this statement is incorrect, we need to prove the following. Let P(x) be “x > 1” P(0) false P(0.1) false P(0.2) false … false P(2) false … false

  16. If this statement is correct, we need to prove the following. Let P(x) be “x > 1” P(0) P(0.1) P(0.2) … true P(2) … e.g.6 • Is the following statement true? • x  R+ (x > 1) Consider x = 2 Note that 2  R+ “2 > 1” is true. This statement is true.

  17. e.g.7 (Page 19) • Is the following statement true? • x  R (y  R (y > x)) If this statement is correct, we need to prove the following. There exists a value y such that P(0, y) is true. Let P(x, y) be “ y > x” P(0, 0) P(0, 0.1) true x = 0 true P(0, 0.2) … There exists a value y such that P(0.1, y) is true. P(0.1, 0) true P(0.1, 0.1) true x = 0.1 P(0.1, 0.2) … x = 0.2 true

  18. P(0, 0) P(0, 0.1) P(0, 0.2) … P(0.1, 0) P(0.1, 0.1) P(0.1, 0.2) … e.g.7 (Page 19) • Is the following statement true? • x  R (y  R (y > x)) If this statement is incorrect, we need to prove the following. Let P(x, y) be “ y > x” There doest not exist a value y such that P(0.1, y) is true. x = 0 That is, for each value y  R, P(0.1, y) is false. false false false x = 0.1 false false x = 0.2

  19. If this statement is correct, we need to prove the following. Let P(x, y) be “ y > x” P(0, 0) P(0, 0.1) true x = 0 true P(0, 0.2) … P(0.1, 0) true P(0.1, 0.1) true x = 0.1 P(0.1, 0.2) … x = 0.2 true e.g.7 • Is the following statement true? • x  R (y  R (y > x)) Let y = x + 1 Note that, if x  R, then y  R y = 1 “y > x” is true. y = 1.1 This statement is true. y = 1.2

  20. e.g.8 (Page 19) • Is the following statement true? • x  R ( y  R (y > x)) If this statement is correct, we need to prove the following. Let P(x, y) be “ y > x” true P(0, 0) P(0, 0.1) true true x = 0 P(0, 0.2) true … true P(0.1, 0) true P(0.1, 0.1) true true x = 0.1 P(0.1, 0.2) true … true x = 0.2 true

  21. P(0, 0) P(0, 0.1) P(0, 0.2) … P(0.1, 0) P(0.1, 0.1) P(0.1, 0.2) … e.g.8 • Is the following statement true? • x  R ( y  R (y > x)) If this statement is incorrect, we need to prove the following. Let P(x, y) be “ y > x” x = 0 false false x = 0.1 x = 0.2

  22. If this statement is incorrect, we need to prove the following. Let P(x, y) be “ y > x” P(0, 0) P(0, 0.1) x = 0 P(0, 0.2) … P(0.1, 0) false P(0.1, 0.1) false x = 0.1 P(0.1, 0.2) … x = 0.2 e.g.8 • Is the following statement true? • x  R ( y  R (y > x)) Consider x = 0.1 and y = 0 Note that x  R and y  R “y > x” is false. (i.e., “0 > 0.1” is false) This statement is false.

  23. e.g.9 (Page 19) • Is the following statement true? • x  R ((x  0)   y  R+ (y > x)) If this statement is correct, we need to prove the following. Let P(x, y) be “ y > x” true P(0, 0.1) P(0, 0.2) true true x = 0 P(0, 0.3) true … true P(0.1, 0.1) P(0.1, 0.2) x = 0.1 P(0.1, 0.3) … x = 0.2

  24. e.g.9 • Is the following statement true? • x  R ((x  0)   y  R+ (y > x)) If this statement is incorrect, we need to prove the following. Let P(x, y) be “ y > x” P(0, 0.1) P(0, 0.2) false false x = 0 P(0, 0.3) … P(0.1, 0.1) P(0.1, 0.2) false x = 0.1 P(0.1, 0.3) false … x = 0.2 false

  25. If this statement is correct, we need to prove the following. Let P(x, y) be “ y > x” true P(0, 0.1) P(0, 0.2) true true x = 0 P(0, 0.3) true … true P(0.1, 0.1) P(0.1, 0.2) x = 0.1 P(0.1, 0.3) … x = 0.2 e.g.9 • Is the following statement true? • x  R ((x  0)   y  R+ (y > x)) Let x = 0 Note that y  R+ (i.e., y > 0) “y > x” is true. (i.e., “y > 0” is true) This statement is true.

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