Minimize SOP and POS algebraically?

Question

Is it possible to simplify an SOP (sum of products) or POS (product of sums) expression algebraically? I can only do it through k-maps.

Example: $a'b'c'd' + a'b'c'd + a'b'cd' + a'b'cd + ab'c'd + abc'd' + abc'd + abcd' + abcd$

Answer 1

You can use the distributive law as usual:

$$ (a+b)c = ac + bc $$

or the other way

$$ (ab)+c = (a+c)(b+c) $$

(it might help to temporarily swap $+$ and $\cdot$ if you have trouble "seeing" the above distribution)

Doing it "algebraically" is unlikely to be any better than Karnaugh maps, or more generally the Quine-McCluskey algorithm. In fact, it will probably be much more work.