logic - how to convert this formula

Question

I have this formula: $$(X \wedge (Y \rightarrow Z)) \vee \neg(\neg X \rightarrow (Y \rightarrow Z))$$ Is it possible to convert it to this: $$X ↔ (Y → Z)$$ the truth table show that they are equivalent. Any clues or hints are appreciated also.

Answer 1

Yes, they are equivalent.

Remember that $P \leftrightarrow Q$ is equivalent to $(P \land Q) \lor (\neg P \land \neg Q)$ ($P \leftrightarrow Q$ means that they're either both true or both false)

So, see if you can convert the first statement to $(X \land (Y \rightarrow Z)) \lor (\neg X \land \neg (Y \rightarrow Z))$

HINT

Use:

Implication

$P \rightarrow Q \Leftrightarrow \neg P \lor Q$

and use DeMorgan

Answer 2

We have that \begin{align}(X \land (Y \rightarrow Z)) \vee \neg(\neg X \rightarrow (Y \rightarrow Z))&=(X\land(\lnot Y\lor Z))\lor\lnot(X\lor(Y\to Z))\\ &=(X\land(\lnot Y\lor Z))\lor\lnot(X\lor(\lnot Y\lor Z)\\ &=(X\land(\lnot Y\lor Z))\lor(\lnot X\land( Y\land \lnot Z))\\ \end{align}

The other is \begin{align}X\leftrightarrow (Y\to Z)&=\lnot((X\lor(\lnot Y\lor Z))\land\lnot(X\land(\lnot Y\lor Z)))\\ &=(\lnot X\land (Y\land \lnot Z)\lor(X\land(\lnot Y\lor Z))\end{align} so indeed the two are equivalent.