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

2. 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 P(1)true P(2)true P(3)true P(4)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. …true

3. 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)true P(1)true P(2)true P(3)true P(4)true … false There may exist another non-negative integer k such that P(k) is false

4. 4 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 P(1)false P(2)true P(3)true P(4)true … m 2 > minteger 0, 1, 2, … -1, -2, … 0 2 > 0 1 2 > 1 2 2 > 2

5. 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) P(2)true P(3) P(4) 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. …

6. 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 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(0)false P(1)false P(2)false P(3)false P(4)false …

7. 7 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 true 2 2 > 2 P(0) P(1) P(2) P(3) P(4) … m 2 > m integer

8. 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. 9 e.g.3 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. Since m is non-negative and n is a positive integer, we derive that q and r are also non-negative. 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. Let Z + be the set of positive integers. Let N be the set of non-negative integers.  n  Z + ( )  m  N ( )  q  N ( )  r  N ( ) (r < n)  (m = qn + r)

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

11. 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. P(0)true P(0.1)true P(0.2)true … P(1)true … Let P(x) be “ x > 1 ”

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

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

14. 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. 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 … P(2)false …

16. 16 e.g.6 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) … Consider x = 2 Note that 2  R + “ 2 > 1 ” is true. This statement is true.

17. 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. Let P(x, y) be “ y > x ” P(0, 0) P(0, 0.1) P(0, 0.2) true … P(0.1, 0) P(0.1, 0.1) P(0.1, 0.2) … true x = 0 x = 0.1 x = 0.2true There exists a value y such that P(0, y) is true. true There exists a value y such that P(0.1, y) is true.

18. 18 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 ” 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) … false x = 0 x = 0.1 x = 0.2 There doest not exist a value y such that P(0.1, y) is true. That is, for each value y  R, P(0.1, y) is false. false

19. 19 e.g.7 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 ” P(0, 0) P(0, 0.1) P(0, 0.2) true … P(0.1, 0) P(0.1, 0.1) P(0.1, 0.2) … true x = 0 x = 0.1 x = 0.2true Let y = x + 1 Note that, if x  R, then y  R “ y > x ” is true. This statement is true. y = 1 y = 1.1 y = 1.2

20. 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 ” P(0, 0) P(0, 0.1) P(0, 0.2) true … P(0.1, 0) P(0.1, 0.1) P(0.1, 0.2) … true x = 0 x = 0.1 x = 0.2true

21. 21 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 ” 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) … false x = 0 x = 0.1 x = 0.2 false

22. 22 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 ” 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) … false x = 0 x = 0.1 x = 0.2 false 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. 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 ” P(0, 0.1) P(0, 0.2) P(0, 0.3) true … P(0.1, 0.1) P(0.1, 0.2) P(0.1, 0.3) … x = 0 x = 0.1 x = 0.2 true

24. 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) P(0, 0.3) false … P(0.1, 0.1) P(0.1, 0.2) P(0.1, 0.3) … x = 0 x = 0.1 x = 0.2 false

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