How do you simplify a boolean algebra expression when it's a two by three expression?

Question

I'm a little stuck in my simplifying of this boolean logic expression. If it was $2 \times 2$, I know I could foil, but I can't find any law that will help me go any further. Would someone help me figure out where to go now?

In this attempt, $\cdot$ stands for logical AND, $+$ for logical OR, and $\overline{A}$ stands for NOT A.

Simplify: $\overline{(A+B)}\cdot\overline{(C+D+E)}+\overline{(A+B)}$ \begin{align} &\overline{(A+B)}\cdot\overline{(C+D+E)}+\overline{(A+B)}\\ \text{de Morgan's law}~~~&(\overline{A}\cdot\overline{B})\cdot(\overline{C}\cdot\overline{D}\cdot\overline{E})+\overline{(A+B)}\\ \text{de Morgan's law}~~~&(\overline{A}\cdot\overline{B})\cdot(\overline{C}\cdot\overline{D}\cdot\overline{E})+(\overline{A}\cdot \overline{B}) \end{align}

Here is an image of my attempt on paper.

Please use MathJax to write math correctly. Also, what operation is in between $\neg(A+B)$ and $\neg(C+D+E)$? — manooooh, Aug 31 '20 at 22:46
When you refer to the expression been a 2 by 2, are you referring to it been solved on a Karnaugh map? — user400188, Sep 01 '20 at 02:07
I mean 2 x 2 by saying another way of writing something like (A + B)(C + D). — Threshold19, Sep 01 '20 at 02:10
Maybe I’m reading it wrong, but it looks to simplify to $\overline{A+B}$. — Michael Burr, Sep 01 '20 at 02:34
Why can’t you foil? Rewrite $C+D+E$ as $C+(D+E)$ and apply your rule. — Michael Burr, Sep 01 '20 at 02:35
Michael Burr, you're right. I can do that according to associative law. But how can you already see the final simplified form? — Threshold19, Sep 01 '20 at 02:47

amrsa · Answer 1 · 2020-09-01T10:56:08.273

1

You can just follow Eric Towers' comment:
Take $U = \overline{A+B}$ and $V = \overline{C+D+E}$, and apply one of the absorption laws: $$(U\cdot V)+U=U.$$ (The other would be $(U+V)\cdot U = U$.)
This was certainly what Michael Burr meant in his first comment.

edited Sep 01 '20 at 10:56

answered Sep 01 '20 at 10:42

amrsa

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How do you simplify a boolean algebra expression when it's a two by three expression?

1 Answers1