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

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)

3. 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.

4. 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?

5. 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

6. 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.

7. 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?

8. 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.

9. 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.

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

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

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

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

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

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