I need to simplify the following expression:
$$ P = AC + A\bar{B} + \bar{A}BC + \bar{A}\bar{B}\bar{C} $$
Using a K-Map I get the correct answer of:
$$ P =AC + A\bar{B} + BC + \bar{B}\bar{C} $$
My problem is that I don't understand how
$$ \bar{A}BC + \bar{A}\bar{B}\bar{C} = BC + \bar{B}\bar{C} $$
I've tried using logical adjacency, expanding variables, DeMorgan's law, and the basic properties like distributivity to manipulate the variables, but I can never end up with the answer I get for the K-Map. Am I missing some theorem or property?
Here's what's really going on: $$ AC + A \bar B + \bar A BC + \bar A \bar B \bar C = \\ AC + A(B + \bar B)C + A \bar B + A \bar B(C + \bar C) + \bar A BC + \bar A \bar B \bar C = \\ AC + A \bar B + [\bar ABC + A BC] + [A \bar B \bar C + \bar A \bar B \bar C] =\\ AC + A \bar B + [\bar A + A]BC + [A + \bar A]\bar B \bar C =\\ AC + A \bar B + BC + \bar B \bar C $$ what we're implicitly doing with the K-map is introducing redundant terms in the sum in order to allow for a tidy factorization.