Question

Write the negation of the following statements:
(i) p : For every positive real number x, the number (x − 1) is also positive.
(ii) q : For every real number x, either x > 1 or x < 1.
(iii) r : There exists a number x such that 0 < x < 1.

Answer 1

(i) ​p: For every positive real number x, the number (x − 1) is also positive.
~p: At least for one positive real number x, the number $(x-1)$ is not positive.

(ii) q: For every real number x, either x > 1 or x < 1.
~q: At least for one real number x, neither $x>1$ nor $x<1$.

(iii) r: There exists a number x such that 0 < x < 1.
~r: For every real number x, either $x\le 0$ or $x<1$.