Boolean Arithmetic Simplification

[BraedenP]

Homework Statement

I am asked to prove that $(\sim x)\vee z = \sim(x\vee y)\vee\sim(y\vee\sim z)\vee\sim(x\vee\sim y)\vee\sim(\sim y\vee\sim z)$.

I've tried using all combinations of DeMoran's rule, the distributive rule to get the y terms together, and the absorption rule to get rid of the y (which is required in order to simplify it down in terms of x and z.

Homework Equations

DeMorgan's Rule: $\sim(p\wedge q) = \sim p\vee\sim q$
Absorption Rule: $p\vee(p\wedge q) = p$

The Attempt at a Solution

I can post some of the steps I've taken, but none really lead anywhere. Where is a good place to start for a question like this?

 

[NascentOxygen]

Convert the OR's contained within brackets to AND's using DeMorgan's rules.

 

[BraedenP]

Thanks -- yeah, I was trying that before, but couldn't get it simplified down enough.

I got the solution now (and it wasn't too hard). I just had to use DeMorgan's and the distributive rule to get rid of all the Ys first, and then everything else just fell into place without much effort.