Check whether the following pair of statement | vedclass

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(A) Let $P$ be the statement: 'There exists real numbers $x$ and $y$ for which $x+y=y+x.$'This is an existential statement of the form 'There exists $

Vedclass · Accepted Answer

Yes,they are negations.

Vedclass · Answer

(A) Let $P$ be the statement: 'There exists real numbers $x$ and $y$ for which $x+y=y+x.$'This is an existential statement of the form 'There exists $x, y$ such that $P(x, y)$'.The negation of an existential statement 'There exists $x$ such that $P(x)$' is 'For all $x$,not $P(x)$'.Therefore,the negation of $P$ is: 'For all real numbers $x$ and $y$,$x+y 
eq y+x.$'Since statement $II$ is exactly this negation,the given pair of statements are indeed negations of each other.

Check whether the following pair of statements are negation of each other. Give reasons for your answer.
$I$: There exists real numbers $x$ and $y$ for which $x+y=y+x.$
$II$: For all real numbers $x$ and $y$,$x+y \neq y+x.$

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Check whether the following pair of statements are negation of each other. Give reasons for your answer.$I$: There exists real numbers $x$ and $y$ for which $x+y=y+x.$$II$: For all real numbers $x$ and $y$,$x+y \neq y+x.$