Boolean Algebra Transform

Question

I am revisiting Boolean algebra after a long while.

Can somebody help show me how to simplify the LHS to get the RHS?

$$abc * a'bc + (abc)' * (a'bc)'\quad = \quad \;b'+c'$$

Answer 1

$$\color{blue}{\bf abc * a'bc} + (abc)' * (a'bc)'$$

Note: $$\color{blue}{\bf abc*a'bc} = abca'bc = (aa')bbcc = F*bc = \color{blue}{\bf F}$$

So we simplify what remains: $$\color{blue}{\bf F} + (abc)' * (a'bc)' = (abc)' * (a'bc)'$$ $$ = (a'+b'+c')*(a + b' + c')\tag{ by Demorgan's.}$$

$$ = (b' + c')+ (a' * a)\tag{Distributive law}$$ $$ = b' + c' + F $$ $$= b' + c'$$