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Statements Containing Multiple Quantifiers

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

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Statements Containing Multiple Quantifiers

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  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 xmust not depend on y, while the choice of zmay 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) = ax2 + bx + c has an x-intercept. • Its negation is a, b, c R, x R, f(x) = ax2 + 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, x2 + bx + c = (x – s)2 + t. • Its negation is b, c R, s, t R, x2 + bx + c (x – s)2 + t. • Which statement is true?

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