Mathematics for Computer Science

Chapter 3 Logical Formulas72

2. 8 x 9 y: x2yD 0


3. 8 x:x < 10IMPLIES. 8 y: y < xIMPLIESy < 9/


4. 8 x 9 y: Œy > x^9z: yCzD100ç

determine which propositions are true when the variables range over:

(a)the nonnegative integers.

(b)the integers.

(c)the real numbers.

Problem 3.20.
LetQ.x;y/be the statement

“xhas been a contestant on television showy.”

The universe of discourse forxis the set of all students at your school and foryis
the set of all quiz shows that have ever been on television.
Determine whether or not each of the following expressions is logically equiva-
lent to the sentence:

“No student at your school has ever been a contestant on a television quiz show.”

(a) 8 x 8 y:NOT.Q.x;y//

(b) 9 x 9 y: NOT.Q.x;y//

(c) NOT. 8 x 8 y: Q.x;y//

(d) NOT. 9 x 9 y: Q.x;y//

Problem 3.21.
Find a counter model showing the following is not valid.

9 x:P.x/IMPLIES 8 x:P.x/

(Just define your counter model. You do not need to verify that it is correct.)