boolean expressions simplification Help needed.

Question

I am stuck simplifying. Can anyone help?

It states that

$$ (XY’+YZ)’ = X’Y’ + X’Z’+YZ’ $$

I tried all axioms yet I can't figure it out.

Answer 1

Notice that: \begin{align*} (XY’+YZ)’ &= (X' + Y)(Y' + Z') & \text{by DeMorgan's Law}\\ &= X'(Y' + Z') + Y(Y' + Z') & \text{by Distributive Law}\\ &= X'Y' + X'Z' + YY' + YZ' & \text{by Distributive Law}\\ &= X'Y' + X'Z' + 0 + YZ' & \text{by Inverse Law}\\ &= X'Y' + X'Z' + YZ' & \text{by Identity Law}\\ \end{align*}

Answer 2

Your first expression $(XY'+YZ)'$ simplifies as $(XY')'(YZ)'=(X'+Y)(Y'+Z')=(X'+Y)Y'+(X'+Y)Z'$.

Then since $YY'=0$, this is $X'Y'+0+X'Z'+YZ'=\boxed{X'Y'+X'Z'+YZ'}$ as desired.