Law on common identities Algebra

Question

I'm studying the theory of algebra and the basic laws. I know the law on common identities

$A+(\bar{A}B)=A+B$

but i wasn't able to demonstrate that :

$\bar{C}+(AB)C=\bar{C}+AB$

this last equation has a similar form of the former equation, but i couldnt succed in demonstrating. Can you help me ?

Answer 1

Yes, the law of Boolean Algebra you are talking about is the absorption law. Note that $$A+ \overline{A}(B)= A +B $$ $$ \overline{C}+\overline{\overline{C}}(AB)=\overline C + C(AB)= \overline{C}+AB$$

Note that in the second step of our second expression, we have tried to express it like the first one, and noting that $\overline{\overline C}= C$, just compare the forms of the two equations and the result follows.