1. Statements Containing Multiple Quantifiers Lecture 11 Section 2.3 Mon, Feb 5, 2007

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

3. Mixed Quantifiers 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.

4. Mixed Quantifiers Which of the following are true?  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.

5. Multiply Quantified Statements In the statement  x  R,  y  R,  z  R, x(y + z) = 0. the predicate x(y + z) = 0 must be true for every y and for some x and for some z. However, the choice of x must not depend on y, while the choice of z may depend on y.

6. Examples Which of the following statements are true?  x  N,  y  N, y < x.  x  Q,  y  Q, y < x.  x  R,  y  R, y < x.  x  Q,  y  Q,  z  Q, x < z < y.

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

8. 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)

9. 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)

10. 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)

11. 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)

12. 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

13. Negation of Multiply Quantified Statements Consider the statement  c  R,  x  R, f(x) = cx has an x-intercept. Its negation is  c  R,  x  R, f(x) = cx has no x-intercept. Which statement is true? How would you prove it?

14. Negation of Multiply Quantified Statements Consider the statement  m, b  R,  x  R, f(x) = mx + b has an x-intercept. Its negation is  m, b  R,  x  R, f(x) = mx + b has no x-intercept. Which statement is true?

15. Negation of Multiply Quantified Statements Consider the statement  a, b, c  R,  x  R, f(x) = ax 2 + bx + c has an x-intercept. Its negation is  a, b, c  R,  x  R, f(x) = ax 2 + bx + c has no x-intercept. Which statement is true?

16. Negation of Multiply Quantified Statements Consider the statement  b, c  R,  s, t  R, x 2 + bx + c = (x – s) 2 + t. Its negation is  b, c  R,  s, t  R, x 2 + bx + c  (x – s) 2 + t. Which statement is true?