Check whether the following pair of statement | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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 $y$,$x+y \neq y+x.$

Similar Questions

Write the negation of the following statement:
$p:$ For every real number $x, x^{2} > x.$

The negation of the statement $\forall x \in N, x^2+x$ is an even number is

Consider the following statements:
$P$: $I$ have fever
$Q$: $I$ will take medicine
$R$: $I$ will take rest
The statement "If $I$ have fever,then $I$ will take medicine and $I$ will take rest" is equivalent to:

The negation of $(p \land q) \to ((p \lor r) \to \sim q)$ is equivalent to ...

The output of the following circuit is

Vedclass Test Series

Exam Paper Generator

Online Exam Module

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 $y$,$x+y \neq y+x.$