Question

Are the following pairs of statements negations of each other?
(i) The number $x$ is not a rational number
The number $x$ is not an irrational number
(ii) The number $x$ is a rational number
The number $x$ is an irrational number

Answer 1

(i) Let $p:$ the number $x$ is not a rational number
The negation of $p$ is "the number $x$ is a rational number"
This is same as the second statement as if a number is not an irrational number then it is a rational number
Therefore the given statements are negations of each other
(ii) Let $p:$ the number $x$ is a rational number
The negation of the first statement is "the number $x$ is not a rational number" .
This means that the number $x$ is an irrational number which is the same as the second statement
Therefore the given statements are negations of each other