Boolean Algebra expanding using absorption

Question

Hi I have a question regarding the absorption law. I was told that I cannot expand ab = ab + abc by writing ab = ab(1+c). However, I believe you can expand xy = xyz' + xyz by doing xy = xy(z' + z) . Why does it work in this case and not the other one?

Answer 1

You can expand.

This is called 'the distributivity law', and it holds for meet over join, as well as for join over meet.

That said, you not only can expand $$ab=ab1=ab(1+c)=ab+abc$$ but you can also write things like $$a+b=a+b+0=a+b+(0c)=(a+b)(a+b+c)\,.$$