Electronic – Optimizing a boolean expression to (NAND,AND,NOT) system

boolean-algebrakarnaugh mapnand

I'm trying to simplify the expression given by the following Karnaugh map to an expression that is using only NANDs, NOTs and ANDs- the less gates (of any kind), the better.

Karnaugh Map

I know how to optimize it to an (OR,AND,NOT) system, but that doesn't seem to help.

I also know how to create an OR gate using only NANDs and NOTs, but that creates a really complicated expression and I've been hinted there's a simple one.

I tried using Wolfram Alpha, but no (AND,NAND,NOT) system exist. The closest one is a (NAND, NOT) system and that expression still looks complicated.

I'm allowed 4 inputs per NAND and 2 inputs per AND.

Thanks in advance!

Best Answer

Complement Law says \$\overline{\overline{X}} = X\$.

$$\overline A\ \overline B\ \overline C + \overline A\ B\ C + A\ B\ D + A\ \overline B\ \overline D$$

Take Double Complement. $$\overline{\overline{\overline A\ \overline B\ \overline C + \overline A\ B\ C + A\ B\ D + A\ \overline B\ \overline D}}$$

Use DeMorgan's to remove lower complement. $$\overline{\overline{\overline A\ \overline B\ \overline C} ∙ \overline{\overline A\ B\ C} ∙ \overline{A\ B\ D} ∙ \overline{A\ \overline B\ \overline D}}$$

4 NOTs, 4 3-input NANDs, 1 4-input NAND.

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