Simplifying this boolean algebra

Question

Can someone help me with simplifying this Boolean algebra

Prove: (x * y') + (y' * z) + (x' * z) = (x * y') + (x' * z)

x'= not x

Can you please show me step by step and the laws which you have applied in proving?

Thank you very much!

Answer 1

It's worth your while to study this proof because this pattern shows up a lot. In fact, it's the proof of an identity known as Consensus:

\begin{array}{l} & xy' + y'z + x'z & \text{ Given }\\ & xy' + y'z1 + x'z & \text{ Identity }\\ & xy' + y'z(x + x') + x'z & \text{ Identity }\\ & xy' + y'zx + y'zx' + x'z & \text{ Distributive }\\ & xy' + y'zx + x'z + y'zx' & \text{ Associative }\\ & xy'(1 + z) + x'z(1+ y') & \text{ Distributive }\\ & xy'1 + x'z1 & \text{ Identity }\\ & xy' + x'z & \text{ Identity }\\ \end{array}

Answer 2

The Consensus Theorem does this in 1 step:

Consensus Theorem

$XY'+Y'Z+X'Z \Leftrightarrow XY'+X'Z$

We can derive the Consensus Theorem (and thus your equivalence as well) from some more basic principles:

$XY'+Y'Z+X'Z= \text{ Adjacency}$ (Adjacency says $PQ+PQ'=P$)

$XY'+XY'Z+X'Y'Z+X'Z= \text{ Absorption}$ ($XY'$ absorbs $XY'Z$ and $X'Z$ absorbs $X'Y'Z$)

$XY+X'Z$

If you didn't already know of Absorption and Adjacency, then you should immediately add them to your Boolean algebra toolbox, because they are super useful!