Simplify this boole algebra expression

Question

Can someone help me with this expression, and how to simplify that with all steps? Im kind of lost in those, and i have exam in 2 days.

Thank you

Expression

The result of expression is: F = B + C * A'

Answer 1

I assume that you can use the usual axioms of Boolean algebras
(that these are bounded complemented distributive lattices).
Let me use $A'$ for your $\bar{A}$ (it's simpler in MathJax).

Now maybe you know, or are able to proof, that for for every $A,B$ in some Boolean algebra

1. $A + AB = A$;
2. using the above equality, you can also show that $A + A'B = A + B$.

And then, of course, as amWhy pointed in a comment, $A + A' = 1$ (this is also one of my original assumptions), and using several laws (starting with distributivity), you can show that $$(A+B')(A'+B) = AB + A'B'.$$

Now \begin{align} (A+A')B + ABC' + (A+ B')(A' + B)C &= B + ABC' +(AB + A'B')C\\ &= B + AB(C+C') + A'B'C\\ &= B + AB + A'B'C\\ &= B + A'B'C\\ &= B + A'C, \end{align} where in the last two equalities I used the results numbered 1. and 2. above.