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

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 $\neg(C+D+E)$? Aug 31 '20 at 22:46
• $U \wedge V \vee U = U$. Aug 31 '20 at 22:58
• It's an and symbol. I'll look into MathJax now. Thank you. Sep 1 '20 at 1:48
• When you refer to the expression been a 2 by 2, are you referring to it been solved on a Karnaugh map? Sep 1 '20 at 2:07
• I mean 2 x 2 by saying another way of writing something like (A + B)(C + D). Sep 1 '20 at 2:10

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 $$(U+V)\cdot U = U$$.)