1. Lecture 7 – Jan 28, 2002

2. Chapter 2 The Logic of Quantified Statements

3. Section 2.1 Predicates and Quantified Statements I

4. Predicates A predicate is a sentence that contains a finite number of variables, and becomes a statement when values are substituted for the variables. “x flies like a y.” Let x be “time” and y be “arrow.” Let x be “fruit” and y be “banana.”

5. Domains of Predicate Variables The domain D of a predicate variable x is the set of all values that x may take on. Let P(x) be the predicate. x is a free variable. The truth set of P(x) is the set of all values of x  D for which P(x) is true.

6. The Universal Quantifier The symbol  is the universal quantifier. The statement  x  S, P(x) means “for all x in S, P(x),” where S  D. x is a bound variable, bound by the quantifier . The statement is true if P(x) is true for all x in S. The statement is false if P(x) is false for at least one x in S.

7. Examples Statement “7 is a prime number” is true. Predicate “x is a prime number” is neither true nor false. Statements “  x  {2, 3, 5, 7}, x is a prime number” is true. “  x  {2, 3, 6, 7}, x is a prime number” is false.

8. Examples of Universal Statements  x  {1, …, 10}, x 2 > 0.  x  {1, …, 10}, x 2 > 100.  x  R, x 3 – x  0.  x  R,  y  R, x 2 + xy + y 2  0.  x  , x 2 > 100.

9. The Existential Quantifier The symbol  is the existential quantifier. The statement  x  S, P(x) means “there exists x in S such that P(x),” S  D. x is a bound variable, bound by the quantifier . The statement is true if P(x) is true for at least one x in S. The statement is false if P(x) is false for all x in S.

10. Examples of Universal Statements  x  {1, …, 10}, x 2 > 0.  x  {1, …, 10}, x 2 > 100.  x  R, x 3 – x  0.  x  R,  y  R, x 2 + xy + y 2  0.  x  , x 2 > 100.

11. Negations of Universal Statements The negation of  x  S, P(x) is the statement  x  S,  P(x). If “  x  R, x 2 > 10” is false, then “  x  R, x 2  10” is true.

12. Negations of Existential Statements The negation of  x  S, P(x) is the statement  x  S,  P(x). If “  x  R, x 2 < 0” is false, then “  x  R, x 2  0” is true.

13. Example: Negation of a Universal Statement p = “Everybody likes me.” Express p as  x  {all people}, x likes me.  p is the statement  x  {all people}, x does not like me.  p = “Somebody does not like me.”

14. Example: Negation of an Existential Statement p = “Somebody likes me.” Express p as  x  {all people}, x likes me.  p is the statement  x  {all people}, x does not like me.  p = “Everyone does not like me.”  p = “Nobody likes me.”

15. Lecture 8 – Jan 29, 2002

16. Section 2.2 Predicates and Quantified Statements II

17. Multiply Quantified Statements Multiple universal statements  x  S,  y  T, P(x, y) The order does not matter. Multiple existential statements  x  S,  y  T, P(x, y) The order does not matter.

18. Multiply Quantified Statements Mixed universal and existential statements  x  S,  y  T, P(x, y)  y  T,  x  S, P(x, y) The order does matter. What is the difference? Compare  x  R,  y  R, x + y = 0.  y  R,  x  R, x + y = 0.

19. Negation of Multiply Quantified Statements Negate the statement  x  R,  y  R,  z  R, x + y + z = 0.  (  x  R,  y  R,  z  R, x + y + z = 0)   x  R,  (  y  R,  z  R, x + y + z = 0)   x  R,  y  R,  (  z  R, x + y + z = 0)   x  R,  y  R,  z  R,  (x + y + z = 0)   x  R,  y  R,  z  R, x + y + z  0

20. Negate the statement “Every integer can be written as the sum of three squares.”  (  n  Z,  r, s, t  Z, n = r 2 + s 2 + t 2 ).  n  Z,  (  r, s, t  Z, n = r 2 + s 2 + t 2 ).  n  Z,  r, s, t  Z,  (n = r 2 + s 2 + t 2 ).  n  Z,  r, s, t  Z, n  r 2 + s 2 + t 2. Is the original statement true?

21. Universal Conditional Statements A universal conditional statement is of the form  x  S, P(x)  Q(x). The converse is  x  S, Q(x)  P(x). The inverse is  x  S,  P(x)   Q(x). The contrapositive is  x  S,  Q(x)   P(x).

22. Negation of Universal Conditional Statements Negate the statement  x  R, x < 10  x 2 < 100.  (  x  R, x < 10  x 2 < 100)   x  R,  (x < 10  x 2 < 100)   x  R, (x < 10)  (x 2  100). Which one is true?

23. Putnam Question A-2 (1981) Two distinct squares of the 8 by 8 chessboard C are said to be adjacent if they have a vertex or side in common. Also, g is called a C-gap if for every numbering of the squares of C with all the integers 1, 2, …, 64, there exist two adjacent squares whose numbers differ by at least g. Determine the largest C-gap g.

24. Putnam Question A-2 (1981) Consider the standard numbering 12345678 910111213141516 1718192021222324 2526272829303132 3334353637383940 4142434445464748 4950515253545556 5758596061626364 Note that the largest difference is 9.

25. Putnam Question A-2 (1981) Could the answer be 9? 9 is the largest C-gap if 9 is a C-gap, and 10 is not a C-gap.

26. Putnam Question A-2 (1981) 10 is not a C-gap if There exists a numbering of the squares such that no two adjacent squares differ by at least 10. Equivalently, there exists a numbering of the squares such that every two adjacent squares differ by at most 9. We have just seen that this is true. Therefore, 10 is not a C-gap.

27. Putnam Question A-2 (1981) Is 9 a C-gap? Consider the two squares that are labeled #1 and #64. There is a path of at most 8 squares linking square #1 and square #64. Of the 7 differences along this path, one must be at least 9, since the total difference is 63. Therefore, 9 is a C-gap.