3

$$(p \lor q) \land (q \implies p) \equiv p$$

Struggling to solve the above with Semantic Equivalence

step 1: implication $( p \lor q ) \land ((\lnot q) \lor p)$

step 2: distributivity $((p \lor q ) \land \lnot q) \lor ((p \lor q) \land p)$

step 3: absorption. $((p \lor q ) \land \lnot q)) \lor p$

step 4: distributivity $((\lnot q \land p) \land (\lnot q \land q)) \lor p )$

step 5: absorption $((\lnot q \land q) \land p )$

step 6: negation $p \lor T$

Not sure where to go from here? Can $p \lor T$ be resolved to $p$?

Robert Z
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Gooze_Berry
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  • $p \lor T \equiv T$. But you made a mistake in the use of the distributive property between step 3 and step 4. – amWhy Apr 23 '20 at 16:34

1 Answers1

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Step 4 should be $$\begin{align}(4) \quad &\equiv ((p \land \lnot q) \lor \underbrace{(q\land \lnot q)}_{\text{F}}) \lor p\tag{distrib.}\\ \\ (5)\quad &\equiv (p \land \lnot q) \lor p \\ \\(6)\quad &\equiv p\end{align}$$

amWhy
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