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 : All cats scratch
(iii) r : For every real number $x$, either $x>1$ or $x<1$
(iv) s : There exist a number $x$ such that $0<x<1$

Answer 1

(i) The negation of statement $p$ is as follows
There exists a positive real number $x$ such that $x-1$ is not positive
(ii) The negation of statement $q$ is as follows
There exists a cat that does not scratch
(iii) The negation of statement $r$ is as follows
There exists a real number $x$ such that neither $x>1$ nor $x<1$
(iv) The negation of statement s is as follows
There does not exist a number $x$ such that $0<x<1$