The negation of the statement P: "For every r | vedclass

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(C) The given statement $P$ is of the form: "For every $x$,$Q(x)$ or $R(x)$",where $Q(x)$ is $x > 5$ and $R(x)$ is $x < 5$.To find the negation of a s

Vedclass · Accepted Answer

There exists a real number $x$ such that neither $x > 5$ nor $x < 5$

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(C) The given statement $P$ is of the form: "For every $x$,$Q(x)$ or $R(x)$",where $Q(x)$ is $x > 5$ and $R(x)$ is $x To find the negation of a statement involving a universal quantifier ("For every"),we change the quantifier to an existential quantifier ("There exists") and negate the inner statement.The negation of "For every $x$,$Q(x)$ or $R(x)$" is "There exists $x$ such that $\sim(Q(x) \lor R(x))$".Using De Morgan's Law,$\sim(Q(x) \lor R(x))$ is equivalent to $\sim Q(x) \land \sim R(x)$.Here,$\sim(x > 5)$ is $x \leq 5$ and $\sim(x Thus,the negation is: "There exists a real number $x$ such that $x \leq 5$ and $x \geq 5$".This is equivalent to saying: "There exists a real number $x$ such that neither $x > 5$ nor $x Therefore,option $C$ is correct.

The negation of the statement $P$: "For every real number $x$,either $x > 5$ or $x < 5$" is:

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