1 / 25

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.

Download Presentation

Quantifiers

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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.

More Related