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}) $

Answer 1

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 $(\bar x + 1) = 1$