# Rewrite logical formula to one with only '$\to$' and '$\neg$'?

How can I write:

$A \to B ∧ B \to A$

as a statement with only '$\to$' and '$\neg$' ?

• What is $\hat{}$? – John Griffin Sep 10 '17 at 16:14
• @JohnGriffin presumably logical and $\land$ \land – Kenny Lau Sep 10 '17 at 16:14
• @JohnGriffin I edited my answes. Thanks! – Raymond Timmermans Sep 10 '17 at 16:16

We have:

$\lnot (P \to Q) \equiv (P \land \lnot Q)$

and thus:

$\lnot (P \to \lnot Q) \equiv (P \land Q)$.

Thus, we have to rewite the "and" in $(A \to B) \land (B \to A)$ to get:

$\lnot ((A \to B) \to \lnot (B \to A))$

$$\begin{array}{cl} & [A \implies B] \land [B \implies A] \\ =& [A \implies B] \land \neg \neg[B \implies A] \\ =& \neg([A \implies B] \implies \neg[B \implies A]) \\ \end{array}$$

Alternatively:

$$\begin{array}{cl} & [A \implies B] \land [B \implies A] \\ =& [A \land B] \lor [\neg A \land \neg B] \\ =& \neg[\neg A \lor \neg B] \lor [\neg A \land \neg B] \\ =& [\neg A \lor \neg B] \implies [\neg A \land \neg B] \\ =& [A \implies \neg B] \implies \neg [\neg A \implies B] \\ \end{array}$$

• @MauroALLEGRANZA right, corrected. – Kenny Lau Sep 10 '17 at 16:29