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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.
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.
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Solution
(i) p: The number x is not a rational number. q: The number x is not an irrational number. Now ~p: The number x is a rational number. ~q: The number x is a irrational number.
∴ ~p = q and ~q = p
Thus p and q are negations of each other.
(ii) Let p: The number x is a rational number. q: The number x is an irrational number. Now ~p: The number x is not a rational number.
~p: The number x is not an irrational number.
∴ ~p = q and ~q= p
Thus p and q are negations of each other.
(i) p: The number x is not a rational number. q: The number x is not an irrational number. Now ~p: The number x is a rational number. ~q: The number x is a irrational number.
∴ ~p = q and ~q = p
Thus p and q are negations of each other.
(ii) Let p: The number x is a rational number. q: The number x is an irrational number. Now ~p: The number x is not a rational number.
~p: The number x is not an irrational number.
∴ ~p = q and ~q= p
Thus p and q are negations of each other.
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