Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 82 — #90
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Chapter 3
Logical Formulas
Problems for Section 3.6
Practice Problems
Problem 3.23.
For each of the following propositions:
1. 8x 9y: 2x y D 0
2. 8x 9y: x 2y D 0
3. 8x: x < 10 IMPLIES .8y: y < x IMPLIES y < 9/
4. 8x 9y: Œy > x ^ 9z: y C z D 100
determine which propositions are true when the variables range over:
(a) the nonnegative integers.
(b) the integers.
(c) the real numbers.
Problem 3.24.
Let Q.x; y/ be the statement
“x has been a contestant on television show y.”
The universe of discourse for x is the set of all students at your school and for y is
the set of all quiz shows that have ever been on television.
Determine whether or not each of the following expressions is logically equivalent
to the sentence:
“No student at your school has ever been a contestant on a television quiz show.”
(a) 8x 8y: NOT.Q.x; y//
(b) 9x 9y: NOT.Q.x; y//
(c) NOT.8x 8y: Q.x; y//
(d) NOT.9x 9y: Q.x; y//