I am trying to learn logic expression simplification using boolean algebra laws/formulas, but I don't understand it at all. We have this expression: $ (x'\wedge y \wedge z' ) \vee (x' \wedge z) \vee (x \wedge y) $ . In an example solution, we did this:
= $ (x'\wedge y \wedge z' ) \vee (x' \wedge z) \vee (x \wedge y) $
= $ (x'\wedge y \wedge z' ) \vee (x' \wedge y \wedge z) \vee (x' \wedge y' \wedge z ) \vee (x \wedge y) $
= $ (x'\wedge y ) \vee (x' \wedge z ) \vee (x \wedge y) $
= $ y \vee (x' \wedge z ) $
However, I don't understand those steps at all. Could someone explain to me what boolean algebra laws/formulas we did apply at those steps and how? I would be grateful. Thanks!
The following general equivalence principles are used:
Adjacency
$p = (p \land q) \lor (p \land q')$
$p = (p \lor q) \land (p \lor q')$
Idempotence
$p = p \lor p$
$p = p \land p$
Applied to your statement:
$ (x'\land y \land z' ) \lor (x' \land z) \lor (x \land y) \overset{Adjacency: \ x' \land z = (x' \land y \land z) \lor (x' \land y' \land z)}{=}$
$ (x'\land y \land z' ) \lor (x' \land y \land z) \lor (x' \land y' \land z) \lor (x \land y) \overset{Idempotence: \ (x' \land y \land z) = (x' \land y \land z) \lor (x' \land y \land z)}{=}$
$ (x'\land y \land z' ) \lor (x' \land y \land z) \lor (x' \land y \land z) \lor (x' \land y' \land z) \lor (x \land y) \overset{Adjacency: \ (x'\land y \land z' ) \lor (x' \land y \land z) = x' \land y}{=}$
$ (x'\land y) \lor (x' \land y \land z) \lor (x' \land y' \land z) \lor (x \land y) \overset{Adjacency: \ (x' \land y \land z) \lor (x' \land y' \land z) = x' \land z}{=}$
$ (x'\land y) \lor (x' \land z) \lor (x \land y) \overset{Adjacency: \ (x'\land y) \lor (x \land y) = y}{=}$
$y \lor (x' \land z)$