Use Boolean algebra properties to prove the given equality.

Question

Use Boolean algebra properties to prove the given equality.. How do I do this?

$\bar{x}yz + \bar{y} + \bar{z} = \bar{x} + \bar{y} + \bar{z}$

I know

$x + \bar{x}y = x + y$

I also know: $\bar{x}yz + \bar{y} + \bar{z} \equiv \bar{x}yz + \bar{yz}$

so I think I can get $\bar{y}(\bar{x}z + \bar{z})$

which turns to $\bar{y}(\bar{z} + \bar{x}) $

By totality $\bar x = \bar x yz + \bar x\bar yz + \bar x y\bar z+ \bar x\bar y\bar z$, so if only there was some way to absorb the later three terms into $\bar y$ and $\bar z$. — Graham Kemp, Nov 17 '15 at 04:42

score 1 · Answer 1 · answered Nov 17 '15 at 06:15

Your first step should be to use $1 = y\,z + \overline{\,y\,z\,}$

$$\bar x + \bar y +\bar z \;=\; \bar x\,(y\,z + \overline{\,y\, z\,}) + \bar y +\bar z$$

Then use $\overline{\,y\,z\,}=\bar y+\bar z$ and then rearrange the statement so you can use $(\bar x + 1) = 1$

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