Lecture 2 –

1. Lecture 2 –

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 domainD 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 allx in S. • The statement is false if P(x) is false for at leastonex 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}, x2 > 0. • x {1, …, 10}, x2 > 100. • xR, x3 – x 0. • xR, yR, x2 + xy + y2 0. • x, x2 > 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 onex in S. • The statement is false if P(x) is false for allx in S.

10. Examples of Existential Statements • x {1, …, 10}, x2 > 0. • x {1, …, 10}, x2 > 100. • xR, x3 – x 0. • xR, yR, x2 + xy + y2 0. • x, x2 > 100.

11. Negations of Universal Statements • The negation of xS, P(x) is the statement xS, P(x). • If “xR, x2 > 10” is false, then “xR, x2 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, x2 < 0” is false, then “xR, x2 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 • Multiply quantified universal statements • xS, yT, P(x, y) • The order does not matter. • Multiply quantified 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 positive integer can be written as the sum of three squares.” • (nZ, a, b, cZ, n = a2 + b2 + c2). • nZ, (a, b, cZ, n = a2 + b2 + c2). • nZ, a, b, cZ, (n = a2 + b2 + c2). • nZ, a, b, cZ, na2 + b2 + c2. • 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 x2 < 100. • (xR, x < 10 x2 < 100) xR, (x < 10 x2 < 100) xR, (x < 10)  (x2 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 • 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.

28. Putnam Question A-2 (1981) • One possible numbering and path

29. Putnam Question A-2 (1981) • 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.