Prove Logical Equivalence

Question

I've been stuck on this one problem for a couple of days now with no clue on how to complete it. I need to prove the following logical equivalence:

(¬P ∧ ¬R) ∨ (P ∧ ¬Q ∧ ¬R) is equivalent to ¬R ∧ (Q ⇒ ¬(P ∧ ¬R))

If anyone could shed some light on this matter, please..

Try distributing and using the fact that $P\implies Q\iff \sim P \vee Q$. — ncmathsadist, Mar 28 '15 at 22:05
the above and you only have 3 variables so you can just use a truth table. if not that show each implies the other/simplify by laws of logic. should be $\neg P\vee Q$ — miniparser, Mar 28 '15 at 22:44
Paraphrasing ncmathsadist's comment: you can use the fact that $A \to B$ is equivalent to $\neg A \vee B$. — A.P., Mar 28 '15 at 23:32

score 1 · Accepted Answer · answered Mar 29 '15 at 02:41

Working first on the left hand expression gives

$$\begin{align} (\lnot P\land\lnot R)\lor(P\land\lnot Q\land\lnot R) &=\lnot R\land(\lnot P\lor(P\land\lnot Q))\\ &=\lnot R\land((\lnot P\lor P)\land(\lnot P\lor\lnot Q))\\ &=\lnot R\land(\lnot P\lor\lnot Q) \end{align}$$

Working next on the right hand expression gives

$$\begin{align} \lnot R\land(Q\implies\lnot(P\land\lnot R)) &=\lnot R\land(\lnot Q\lor\lnot(P\land\lnot R))\\ &=\lnot R\land(\lnot Q\lor\lnot P\lor R)\\ &=(\lnot R\land(\lnot Q\lor \lnot P))\lor(\lnot R\land R)\\ &=\lnot R\land(\lnot Q\lor \lnot P) \end{align}$$

Prove Logical Equivalence

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