# Proving Logical Equivalence of Multiplexer Implementations

Given the following two propositional forms, prove whether or not they are logically equivalent.

$(a \wedge \neg c) \vee (b \wedge c)$

$(a \wedge \neg c) \vee (b \wedge c) \vee (a \wedge b)$

I've written out a truth table for both expressions, and I can see that they are in fact logically equivalent; however, I was wondering how to go about approaching this problem more elegantly using rules of equivalence as opposed to a brute force truth table solution. Thank you!

• The consensus theorem. – Fabio Somenzi Feb 5 '18 at 19:09
• That clarified things nicely thank you. – rcmpgrc Feb 6 '18 at 1:21

$$(a \land \neg c) \lor (b \land c) \lor (a \land b) \overset{Adjacency}{\Leftrightarrow}$$
$$(a \land \neg c) \lor (b \land c) \lor (a \land b \land c) \lor (a \land b \land \neg c)\overset{Absorption \times 2}{\Leftrightarrow}$$
$$(a \land \neg c) \lor (b \land c)$$