simplify: $AB(C+BD')+(AB)'$ boolean algebra

Question

My solution
$S= AB(C+BD')+(AB)'$
$S= ABC + ABD' + A' + B'$
How do I proceed from this ?

Answer 1

Treat $AB$ as a variable. If it is $0$ the whole expression is true; if it is $1$ the whole expression reduces to $C+BD'$. Thus we may simplify to $$(AB)'+C+BD'=A'+B'+C+BD'$$ By the same logic, $B'+BD'=B'+D'$, so we further reduce to $$A'+B'+C+D'$$