Statements Containing Multiple Quantifiers Lecture 9 Section 2.3 Wed, Jan 26, 2005

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)

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.

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. For those that are false, what is their negation?

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

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.

Negation of Multiply Quantified Statements Consider the statement n  N, r, s, t  N, n = r2 + s2 + t2. Its negation is n  N, r, s, t  N, n  r2 + s2 + t2. Which statement is true? How would you prove it?

Example In the Theory of Computing there is an important theorem: The Pumping Lemma. The Pumping Lemma: For every regular set S, there exists a positive integer n, such that for every word w  S for which |w|  n, there exist words x, y, z, such that for every nonnegative integer k, xykz  S.

Example The Pumping Lemma:  regular set S,  a positive integer n, such that  word w  S for which |w|  n,  words x, y, z, such that  nonnegative integer k, xykz  S.

Example The Pumping Lemma:  S  R,  n  N,  w  S for which |w|  n,  x, y, z *,  k  N, xykz  S.

Example The Pumping Lemma:  S  R,  n  N,  w  S for which |w|  n,  x, y, z *,  k  N, xykz  S.

Example The Pumping Lemma:  S  R,  n  N,  w  S for which |w|  n,  x, y, z *,  k  N, xykz  S.

Example The Pumping Lemma:  S  R,  n  N,  w  S for which |w|  n,  x, y, z *,  k  N, xykz  S.

Example The Pumping Lemma:  S  R,  n  N,  w  S for which |w|  n,  x, y, z *,  k  N, xykz  S.

Example The Pumping Lemma:  S  R,  n  N,  w  S for which |w|  n,  x, y, z *,  k  N, xykz  S.