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?

enter image description here ! my attempt

Have you tried anything yourself? We really don;t like people to just post questions here: we would like to see some effort. And that will also help us point to specific things for you to think about when getting problems like this in the future. So please add your attempts in the post. Thanks! — Bram28, Oct 26 '21 at 17:34

score 1 · Answer 1 · answered Oct 26 '21 at 19:57

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:

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

1 Answers1