Write the negation of the following statement:
$r:$ For every real number $x$,either $x > 1$ or $x < 1.$
$r:$ For every real number $x$,either $x > 1$ or $x < 1.$
(N/A) The negation of a statement involving the quantifier 'for every' is 'there exists'.
The negation of 'either $P$ or $Q$' is 'neither $P$ nor $Q$',which is equivalent to 'not $P$ and not $Q$'.
Therefore,the negation of statement $r$ is:
There exists a real number $x$ such that $x \leq 1$ and $x \geq 1$.
The negation of 'either $P$ or $Q$' is 'neither $P$ nor $Q$',which is equivalent to 'not $P$ and not $Q$'.
Therefore,the negation of statement $r$ is:
There exists a real number $x$ such that $x \leq 1$ and $x \geq 1$.
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The Boolean expression $(p \wedge q) \Rightarrow ((r \wedge q) \wedge p)$ is equivalent to:
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$S_1 \equiv (\sim q \wedge p) \wedge q$
$S_2 \equiv [p \wedge (p$ $\rightarrow q)]$ $\rightarrow q$
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$S_1 \equiv (\sim q \wedge p) \wedge q$
$S_2 \equiv [p \wedge (p$ $\rightarrow q)]$ $\rightarrow q$
$S_3 \equiv (p \wedge q) \wedge (\sim p \vee \sim q)$
$S_4 \equiv (p \wedge q) \rightarrow r$
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Write the following statement in the form "if $p$,then $q$":
$p$: It is necessary to have a password to log on to the server.
$p$: It is necessary to have a password to log on to the server.
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