Question about something I found in a proof

Question

I came across in a proof I was reading in my textbook that

$ab - a'b' = ab - ab' + ab' - a'b'$.

I was wondering why/how that equality is true.

Answer 1

Look at the right side. Notice what's different: $$ab\color{red}{ - ab' + ab' }- a'b'$$

Well, guess what? Those are the same term, one being subtracted by the other. So $- ab' + ab' $ will just yield $0$ and you're left with what you have on the left side.

Answer 2

You'll often see this kind of technique refered to variously as 'adding in zeros' or 'adding a trivial component' - something along these lines. Obviously $-ab'+ab'=0$ so the author has literally added in zero to the right hand side, which does not change the value.