[Math] How to simplify Boolean Expression $\bar B + \bar C (B + A)$

Question

I trying to figure out how $ \bar B + \bar C (B + A)$ simplifies to
$ \bar B + \bar C$.

Answer 1

It's usually easier to figure these things out with Karnaugh maps. After playing around, we see that: \begin{align*} \overline B + \overline C(B + A) &= \overline B(1) + \overline C(B + A) \\ &= \overline B(1 + \overline C) + \overline C(B + A) \\ &= \overline B + \overline B ~ \overline C + \overline C(B + A) \\ &= \overline B + \overline C(\overline B + B + A) \\ &= \overline B + \overline C(1 + A) \\ &= \overline B + \overline C(1) \\ &= \overline B + \overline C \end{align*} as desired.