Write negations for each of the following statements:

Question

(a) For all integers n, if n is prime then n is odd.
(b) ? real numbers x, if x < 1 then 1/x > 1.
(c) For all integers a and b, if a2 divides b2 then a divides b.
(d) ? real numbers x, if x(x ? 2) > 0 then x > 2 or x < 0.
(e) ? real numbers x, if x(x ? 2) ? 0 then 0 ? x ? 2.
(f) For all real numbers x and y with x < y, there exists an integer n such that x ? n ? y.

Answer 1

a. There is an integer n such that n is prime and n is not odd.
b. ? a real number x such that x < 1 and 1/x ? 1:
Or: ? a real number x such that x < 1 and 1/x ? 1:
c. There are integers a and b such that a^2 divides b^2 and a does not divide b:
d. ? a real number x such that x(x ? 2) > 0 and x ? 2 and x ? 0:
Or: ? a real number x such that x(x ? 2) > 0 and 0 ? x ? 2:
e. ? a real number x such that x(x ? 2) > 0 and either 0 > x or x > 2:
Or: ? a real number x such that x(x ? 2) > 0 and either 0 ? x or x ? 2:
f. There are real numbers x and y with x < y such that for all integers n, either x > n or n > y:
Or: There are real numbers x and y with x < y such that for all integers n, either x ? n or n ? y: