Is the boolean expression $(W+Y)Z'=WY'Z'$?

Question

Simplify the boolean expression$WX+XY+X′Z′+WY′Z′$

Using deliberate introduction of $WZ'$ from the Consensus Theorem, $WY'Z'$ can be eliminated using another theorem: $$(Z')W+(Z')WY=Z'W$$

So that it becomes

$$ WX+XY+X'Z' $$

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Usually, simplification using this technique is quite difficult to come up to depending on the expression.

However, I often avoid deliberate introductions and usually use the Consensus Theorem first in simplifications. But I can't seem to solve the problem.

$$ WX+XY+X′Z′+WY′Z′ $$

$$ X(W+Y) + X'Z' + WY'Z' $$ In order to eliminate the last term as done in the first solution, I must have $$ (W+Y)Z'=WY'Z' $$ If this is true, the Consensus Theorem follows and the simplification is done.

Answer 1

If $W=0,Y=1, $ the left is $Z'$ and the right is $0$. If $W=Y=1$ the left is $Z'$ and the right is $0$.

Answer 2

The Consensus Theorem as stated does not allow you to directly eliminate the $WY'Z'$ term, and you indeed need to either first introduce the $WZ$ term, or do a 'reverse absorption' to introduce $X'Y'Z'$ (from $X'Z'$) and $WXZ'$ (from $WX$), combine those with $WY'Z'$ to get $(X'Y' + WX + WY')Z$, and then use the Consensus theorem to eliminate the $WY'$ term (effectively removing the $WY'Z$ term). Moreover, those last few steps can be done in 1 step if we generalize the Consensus Theorem to:

$AXY + AY'Z + AXZ = AXY + AY'Z$ (All terms have a common $A$)

Unfortunately, that 'reverse' Absorption method requires some insight just like your original 'reverse' Consensus does.

So, what may help, is to have yet another generalization of the Consensus Theorem:

$XY + Y'Z + AXZ = XY + Y'Z$ (the 'consensus term' has an extra $A$)

With that, you can go straight from $WX + X'Z' + WY'Z'$ to $WX + X'Z'$ since the consensus term merely has an extra $Y'$

Of course, an excellent to gain insight as to how to combine and/or break up terms is to draw a K-map!