# Among the 4 logical expressions below, determine which are logically equivalent and which are negation of each other.

First question. I noticed my tutor likes to use these two symbols, $\leftrightarrow \Leftrightarrow$ interchangeably. Can I also do so? Are they similar to each other?

Among the 4 logical expressions below, determine which are logically equivalent and which are negation of each other.

(1) $\neg(p\Leftrightarrow q)$

(2) $(\neg p)\Leftrightarrow(\neg q)$

(3) $(\neg p)\Leftrightarrow q$

(4) $p\Leftrightarrow (\neg q)$

(1) $\neg(p\Leftrightarrow q)\equiv \neg((p\to q)\wedge (q\to p))\equiv \neg(p\to q)\vee \neg(q \to p))\equiv\neg(\neg p\vee q)\vee \neg(\neg q \vee p)\equiv (p\wedge\neg q)\vee(q\wedge \neg p)$

(2) $(\neg p)\Leftrightarrow(\neg q)\equiv (\neg p \to \neg q)\wedge(\neg q \to\neg p)\equiv(p\vee\neg q)\wedge(q\vee\neg p)$

So far it seems (1)$\not\equiv$(2), in-fact, when I take the negation of (1), i.e $\neg$(1) , I get $\neg((p\wedge\neg q)\vee(q\wedge \neg p))\equiv\neg(p\wedge\neg q)\wedge\neg(q\wedge\neg p)\equiv(\neg p\vee q)\wedge(\neg q\vee p)\equiv(\neg q\vee p)\wedge(\neg p \wedge q)\equiv(p\vee\neg q)\wedge(q\vee\neg p)$

Therefore, (1) and (2) are negations of each other! Lets continue...

(3) $(\neg p)\Leftrightarrow q\equiv(\neg p \to q)\wedge( q\to\neg p)\equiv(p\vee q)\wedge(\neg q\vee\neg p)$

(4) $p\Leftrightarrow (\neg q)\equiv(p\to\neg q)\wedge(\neg q\to p)\equiv(\neg p\vee\neg q)\wedge(q\vee p)\equiv(p\vee q)\wedge(\neg q\vee\neg p)$

It is evident that (3) and (4) are logically equivalent to each other.

(i), (iii), (iv) are logically equivalent to each other, and are negation to (ii).

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Start from your expression of (1), that is, $(p\wedge\neg q)\vee(q\wedge \neg p)$, and distribute $\wedge$ over $\vee$. This yields that (1) is the conjunction of $\neg q\vee\neg p$, $\neg q\vee q$, $p\vee\neg p$, and $p\vee q$. Two of these are tautologies hence (1) is equivalent to $(\neg q\vee\neg p)\wedge(p\vee q)$, which you showed to be equivalent to your (4). Hence (1) and (4) are equivalent.

The other cases you already proved.

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Excellent~~~~~Thanks. –  Yellow Skies Oct 1 '12 at 13:33