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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 ?

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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.

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  • $\begingroup$ The thing i dont understand is the last passage. $\endgroup$ – Poiera Dec 12 '17 at 8:30
  • $\begingroup$ @Poiera Note that $(AB)$ in the second equation takes the role of the $(B)$ in the first. Also, that, $\overline C$ takes the part of $A$ in the first. $\endgroup$ – Rohan Dec 12 '17 at 8:31
  • $\begingroup$ Yes, but i dont understand why (AB) can take the role of (B). Which axiom or theorem allows this ? $\endgroup$ – Poiera Dec 12 '17 at 8:37
  • $\begingroup$ @Poiera Do you understand why (\overline C) can take the role of (A)? $\endgroup$ – bof Dec 12 '17 at 8:44
  • $\begingroup$ Yes, but no why (AB) can take the role of (B).. In my post, the latter law is a more complex version of the former. I dont know why that equation lead to that result $\endgroup$ – Poiera Dec 12 '17 at 9:34

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