# Rewriting logic sentences

Suppose we have $A$ and $B$, then rewrite the following sentences with only parentheses, sentence letters, negation and disjunction:

$$A \wedge B$$ $$A \rightarrow B$$ $$A \leftrightarrow B$$

~~~

I haven't had much experience dealing with logic, but my attempt for $A \wedge B$ was to rewrite it as $\neg (\neg A \vee \neg B)$, which I shortly realised was incorrect since that was logically equivalent to $A \vee B$.

I haven't the slightest clue on how to approach $A \rightarrow B$, but I would guess that it can be written as $\neg (A \wedge \neg B)$.

• You already rewrote the second one as $\neg (A \wedge \neg B)$. The only thing to deal with is the conjunction. And you know how to rewrite conjunction. – DHMO Apr 19 '17 at 9:35
• Your rewriting for the first one is correct. – DHMO Apr 19 '17 at 9:35

You should familiarize yourself with the De Morgan's laws.

• $A \land B \ \equiv \ (\neg \neg A) \land (\neg \neg B) \ \equiv \ \neg((\neg A) \lor (\neg B))$

• $A \implies B \ \equiv \ \neg(A \land (\neg B)) \ \equiv \ \neg A \lor B$

• $A \iff B \ \equiv \ (A \land B) \lor ((\neg A) \land (\neg B)) \ \equiv \ \neg(\neg A \lor \neg B) \lor \neg(A \lor B)$