How do I simplify this Boolean expression

Question

Number 5 in the following image is what I need help with.

So I had a test a few days ago and I couldn’t simplify this Boolean expression. The answer as shown in the image is supposed to be ~a~bc~d.

Does anyone mind sharing a solution?

! my attempt

Answer 1

The first thing I would note is that even though you have four variables here, you have only two reoccurring terms: $a + b$ and $cd'$.

So, I would first use $P$ for $a + b$, and $Q$ for $cd'$, and then see what I get:

(oh right, I would also strongly recommend using ' instead of overbars ... much easier to work with, both on paper as with mathjax :P )

OK, so let's rewrite $((a+b)'+cd')[cd' \oplus (a+b)]$ as $(P'+Q)(Q \oplus P)$, and do some algebra on that:

$(P'+Q)(Q \oplus P) = (P'+Q)(QP'+Q'P) = P'QP' + P'QP + QQP' + QQ'P = P'Q+0+QP'+0=P'Q$

OK, so plugging $P$ and $Q$ back in, we get:

$PQ' = (a+b)'cd' = a'b'cd'$