How to simplify following boolean expression? A+A'BC+BC'?

Question

please help I'm stuck. I'm trying to solve this.

so far I have:
a) a+b+c
b) a+bc
c) a+b
d) a+b

but for e) I can't progress further since I don't know how to deal with a'bc in this case. anyone so kind to please help me?

Answer 1

$$a + \overline abc = a + bc$$ since we have $a$ or $\overline a bc$. If not $a$, then $\overline a$. That must follow if not a. So "it goes without saying", that if not $a$, (then we already know $\overline a$) so it suffices to assert $bc$.

A similar argument can be made for $bc + \overline bc$, but perhaps simpler, in this case, we can use the distributive property: $$bc + \overline b c = (b +\overline b)c = 1c = c$$

Combining the simplifications gives us $$a+\bar a b c+\bar b c=a+c$$

Answer 2

$a+\bar a b c= a+bc$ and $bc+\bar b c=c$, so $a+\bar a b c+\bar b c=a+c$