Question

Check whether the following pair of statements are negations of each other. Give reasons for your answer.

(i) x +y =y +x is true for every real numbers x and y.

(ii) There exists real numbers x and y for which x + y = y + x.

Answer 1

Let p : x + y = y + x is true for every real numbers x and y.
q: There exists real numbers x and y for which x + y = y + x.

Now ~p: There exists real numbers x and y for which $x+y\ne y+x$. Thus ~$p\ne q.$