Simplifying a Karnaugh Map

Question

In this image it says the Output should be A' + B, but to me, it seems like it should be A' + AB. If I'm wrong, could someone explain to me how A' + B is the correct answer? The reason I ask this is because in the second group of ones, you get ABC + ABC'. Since A and B areboth common in the last two groups, shouldn't they both be included? Thanks in advance.

Karnaugh Map Simplification Example

Answer 1

The key bit you are missing is that A' + B is equal to A' + AB

Because of the Distributive Property of boolean algebra we can rewrite

A' + AB

as the product

(A' + A)(A' + B)

and since the first term is the same as just saying 1, we are left with only our second term.