Are the following pairs of statements negations of each other:
$1$. The number $x$ is not a rational number.
$2$. The number $x$ is not an irrational number.
$1$. The number $x$ is not a rational number.
$2$. The number $x$ is not an irrational number.
(A) The negation of the first statement,'$x$ is not a rational number',is '$x$ is a rational number'.
Since the set of real numbers consists of rational and irrational numbers,the statement '$x$ is not an irrational number' is logically equivalent to '$x$ is a rational number'.
Therefore,the two given statements are negations of each other.
Since the set of real numbers consists of rational and irrational numbers,the statement '$x$ is not an irrational number' is logically equivalent to '$x$ is a rational number'.
Therefore,the two given statements are negations of each other.
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