Check whether the following pair of statement | vedclass

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(A) Let $P$ be the statement: $x+y=y+x$ is true for every real number $x$ and $y$.The negation of a statement of the form 'for all $x$,$P(x)$ is true'

Vedclass · Accepted Answer

Yes,they are negations.

Vedclass · Answer

(A) Let $P$ be the statement: $x+y=y+x$ is true for every real number $x$ and $y$.The negation of a statement of the form 'for all $x$,$P(x)$ is true' is 'there exists at least one $x$ such that $P(x)$ is false'.Therefore,the negation of statement $I$ is: There exists real numbers $x$ and $y$ for which $x+y 
eq y+x$.Comparing this with statement $II$,we see that statement $II$ is indeed the negation of statement $I$.

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 number $x$ and $y$.
$II$: There exists real numbers $x$ and $y$ for which $x+y \neq y+x$.

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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 number $x$ and $y$.$II$: There exists real numbers $x$ and $y$ for which $x+y \neq y+x$.