Quantifiers Supplementary Notes Prepared by Raymond Wong Presented
Quantifiers Supplementary Notes Prepared by Raymond Wong Presented by Raymond Wong 1
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 … 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. 2
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 P(2) true P(3) true P(4) true … true There may exist another non-negative integer k such that P(k) is false 3
e. g. 1 We are going to prove the following claim C is false: statement P(m) 0, 2, 1, … 2, … m 2 > m is true for each non-negative integer m, namely 0, 1, -2, … 02 > 0 P(0) false P(1) false P(2) true P(3) true P(4) true … true 12 > 1 22 > 2 4
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) P(3) true 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(4) … 5
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 P(2) false P(3) false P(4) false … 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. 6
e. g. 2 We are going to prove the following claim C is true: there exists a non-negative m 2 P(m) > m is true integer m such that statement P(0) P(1) P(2) 22 > 2 true P(3) P(4) … 7
e. g. 3 (Page 13) n 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. 8
e. g. 3 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 integers q and r, with 0 r < n such that m = qn + r. 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) )) )) 9
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 If we remove the universe, then we can see the order in which the quantifier occurs n ( m ( q ( r p(m, n, q, r) ) )) 10
e. g. 5 (Page 19) n Is the following statement true? n 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 11
e. g. 5 n Is the following statement true? n x R+ (x > 1) 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) … 12
e. g. 5 n Is the following statement true? n x R+ (x > 1) If this statement is incorrect, we need to prove the following. Let P(x) be “x > 1” P(0) P(0. 1) P(0. 2) Consider x = 0. 1 false Note that 0. 1 R+ “ 0. 1 > 1” is false. … P(1) … This statement is false. 13
e. g. 6 (Page 19) n Is the following statement true? n 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) … 14
e. g. 6 n Is the following statement true? n 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 15
e. g. 6 n Is the following statement true? n x R+ (x > 1) If this statement is correct, we need to prove the following. Let P(x) be “x > 1” P(0) Consider x = 2 P(0. 1) Note that 2 R+ P(0. 2) “ 2 > 1” is true. … P(2) true … This statement is true. 16
e. g. 7 (Page 19) n Is the following statement true? n x R ( y R (y > x)) If this statement is correct, we need to prove the following. Let P(x, y) be “ y > x” x=0 P(0, 0) P(0, 0. 1) P(0, 0. 2) true There exists a value y such that P(0, y) is true … x = 0. 1 x = 0. 2 P(0. 1, 0) P(0. 1, 0. 1) P(0. 1, 0. 2) … true There exists a value y such that P(0. 1, y) is true 17
e. g. 7 (Page 19) n Is the following statement true? n 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 P(0, 0) P(0, 0. 1) There doest not exist a value y such that P(0. 1, y) is true. P(0, 0. 2) That is, for each value y R, P(0. 1, y) is false. … x = 0. 1 x = 0. 2 P(0. 1, 0) false P(0. 1, 0. 1) false P(0. 1, 0. 2) false … false 18
e. g. 7 n Is the following statement true? n x R ( y R (y > x)) If this statement is correct, we need to prove the following. Let P(x, y) be “ y > x” x=0 P(0, 0) P(0, 0. 1) = 1 true P(0, y 0. 2) Let y = x + 1 true … x = 0. 1 x = 0. 2 “y > x” is true. P(0. 1, 0) y =0. 1) 1. 1 true P(0. 1, 0. 2) … Note that, if x R, then y R y = 1. 2 true This statement is true. 19
e. g. 8 (Page 19) n Is the following statement true? n x R ( y R (y > x)) If this statement is correct, we need to prove the following. Let P(x, y) be “ y > x” x=0 x = 0. 1 x = 0. 2 P(0, 0) P(0, 0. 1) true P(0, 0. 2) true … true P(0. 1, 0) true P(0. 1, 0. 1) true P(0. 1, 0. 2) true … true 20
e. g. 8 n Is the following statement true? n 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 P(0, 0) P(0, 0. 1) P(0, 0. 2) … x = 0. 1 x = 0. 2 P(0. 1, 0) false P(0. 1, 0. 1) P(0. 1, 0. 2) … false 21
e. g. 8 n Is the following statement true? n 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 P(0, 0) P(0, 0. 1) Consider x = 0. 1 and y = 0 Note that x R and y R P(0, 0. 2) … x = 0. 1 x = 0. 2 P(0. 1, 0) false P(0. 1, 0. 1) P(0. 1, 0. 2) … “y > x” is false. (i. e. , “ 0 > 0. 1” is false) false This statement is false. 22
e. g. 9 (Page 19) n Is the following statement true? n 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” x=0 x = 0. 1 x = 0. 2 P(0, 0. 1) P(0, 0. 2) true P(0, 0. 3) true … true P(0. 1, 0. 1) P(0. 1, 0. 2) P(0. 1, 0. 3) … 23
e. g. 9 n Is the following statement true? n 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” x=0 P(0, 0. 1) P(0, 0. 2) false P(0, 0. 3) … x = 0. 1 x = 0. 2 P(0. 1, 0. 1) P(0. 1, 0. 2) P(0. 1, 0. 3) false … false 24
e. g. 9 n Is the following statement true? n 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” x=0 x = 0. 1 x = 0. 2 P(0, 0. 1) P(0, 0. 2) true P(0, 0. 3) true … true P(0. 1, 0. 1) P(0. 1, 0. 2) P(0. 1, 0. 3) … true Let x = 0 true Note that y R+ (i. e. , y > 0) “y > x” is true. (i. e. , “y > 0” is true) This statement is true. 25
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