1. Statements Containing Multiple Quantifiers
Lecture 9
Section 2.3
Fri, Feb 3, 2006

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. Mixed Quantifiers Mixed universal and existential statements Compare
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 “There is a woman who is right for every man.”
“For every man, there is a woman who is right for him.”

5. 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?

6. 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)

7. 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)

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)
 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)
 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)
 x  R, y  R, z  R, (x + y + z = 0)
 x  R, y  R, z  R, x + y + z  0

11. 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.

12. 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?

13. Example Write the following statement using quantifiers.
“There exists a computer program that can read the code of any computer program and determine whether that program will eventually halt when it is executed.”

14. Example Write the following statement using quantifiers.
“For every computer program, there exists a computer program that can read its code and determine whether it will eventually halt when it is executed.”

15. Superbowl Predictor
I have a computer program that will correctly predict whether the Seattle Seahawks will win the Superbowl on Sunday.

16. Superbowl Predictor
Example1.cpp
Example2.cpp