Boolean Algebra Missing Step

Question

$A$$\bar{B}$ + $\bar{B}$$C$ + $\bar{A}$$C$ is the equation I am stuck on. It says it can be simplified down to $A$$\bar{B}$ + $\bar{A}$$C$. I can't find any rules that can match this. The steps(and the name of the steps eg. $1$+$A$ = $1$) to solve this would be appreciated.

Answer 1

The rule that matches this is the Consensus Theorem, from where the following algebraic manipulations can be adapted $$\begin{aligned}A\overline{B}+\overline{B}C+\overline{A}C&=A\overline{B} + \overline{A}C+\overline{B}C\\&=A\overline{B} + \overline{A}C+(A+\overline{A})\overline{B}C\\&=A\overline{B}(1+C) + \overline{A}C(1+\overline{B})\\&=A\overline{B} + \overline{A}C \end{aligned}$$

Answer 2

Using the following equivalences:

Absorption

$P + PQ = P$

Adjacency

$PQ + P\overline{Q}=P$

you can do this:

$$A\overline{B}+\overline{B}C+\overline{A}C \overset{Adjacency}{=} $$

$$A\overline{B}+A\overline{B}C+\overline{A}\overline{B}C+\overline{A}C \overset{Absorption \ x \ 2}{=}$$

$$A\overline{B}+\overline{A}C$$