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Given:

$\sim( p \leftrightarrow (q \vee r) )$

$p:$ It's raining

$q:$ The sun is shining

$r:$ There are clouds in the sky.

Translate the proposition into spoken language.

Solution:

It's false that it's raining if and only if the sun is shining or there are clouds in the sky.

Would the solution above be okay?

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    $\begingroup$ Pretty much; I'd add a comma after "that". You might also just say "It's not raining if and only if ...". $\endgroup$ – James Aug 22 '14 at 6:07
  • $\begingroup$ @James Okay, I get it! However, you suggested "It's not raining if and only if..." but isn't the negation symbol affecting the whole proposition? I understand by writing that only $p$ is being negated. Am I wrong? $\endgroup$ – rickHdz Aug 22 '14 at 6:28
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    $\begingroup$ @James: A comma after 'that' would definitely be ungrammatical. $\endgroup$ – TonyK Aug 22 '14 at 8:01
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$\sim(A\leftrightarrow B)\equiv (A\leftrightarrow \sim B)$.

So $\sim(p\leftrightarrow(q\vee r)\equiv p\leftrightarrow (\sim q \wedge \sim r)$

So I'd suggest: It's raining if and only if the sun is not shining and there are no clouds in the sky.

[I wonder if your $r$ should be 'there are no clouds in the sky'. That would make more sense.]

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Your translation is perfectly adequate. Another suggestion:

It's not the case that it's raining if and only if the sun is shining or there are clouds in the sky.

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