How can I transform left side into right side in Logic Propositional?

Question

I have this formulas:

$$(P \to Q) \to (P \land Q) = (\lnot P \to Q) \land (Q \to P), $$

How can I transform left side into right side, or virse-versa, using Logic Propositional?

Answer 1

Hint: Use implication equivalence and distribution to meet in the middle.$$\begin{align} &(P\to Q)\to (P\wedge Q) \\=~& \\=~& \\=~& \\=~& P\wedge\top & \top\text{ is $``$true"} \\=~& P \\=~& P\vee \bot& \bot\text{ is $``$false"} \\=~& \\=~& \\=~& (\neg P\to Q)\wedge(Q\to P)\end{align}$$

Answer 2

Use truth table for both logical expressions $(P\to Q)\to (P\wedge Q)$ and $(\lnot P \to Q) \land (Q \to P)$. $$ \begin{array}{|c|c| c|c| c|} \hline P & Q & (P\to Q) & (P\wedge Q) & (P\to Q)\to (P\wedge Q) \\\hline V & V & V & V & V \\\hline V & F & F & F & V \\\hline F & V & V & F & F \\\hline F & F & V & F & F \\\hline \end{array} $$ and $$ \begin{array}{|c|c| c|c| c|c| c|} \hline P & Q & \lnot P & Q & (\lnot P \to Q) & (Q \to P) & (\lnot P \to Q) \land (Q \to P) \\\hline V & V & F & V & V & V & V \\\hline V & F & F & V & V & V & V \\\hline F & V & V & V & V & F & F \\\hline F & F & V & F & F & V & F \\\hline \end{array} $$