Simplification of sum of products

Question

I have the following equation: A'BC + AB'C + ABC' + ABC

I know I can simplify one part of the equation factoring AB(C' + C) = AB I looked at the results in an online solver and the simplification of the whole function is BC + AC + AB

However I don't understand what properties were used to simplify the other two products.

I was wondering if someone could please provide some insight in how to achieve this result.

Answer 1

So we can simplify as follows

\begin{align*}A'BC + AB'C + ABC' + ABC &= A'BC + AB'C + ABC' + ABC + ABC + ABC\\ &= A'BC + ABC + AB'C + ABC + ABC' + ABC\\ &= BC(A' + A) + A(B'+B)C + AB(C' + C)\\ &= BC + AC + AB. \end{align*}

The main tool we used here was that $ABC = ABC + ABC = ABC + ABC +ABC$.

Let me know if you want me to prove this fact.