[Math] Simplify $ABC+A’BC+AB’C+A’B’C’$ to logic gates

Question

I've simplified $ABC+A'BC+AB'C+A'B'C'$ to $BC+AC+A'B'C'$. However, I want to go further to logic gates for which ICs are readily available. I would like to use at most three such ICs.

Answer 1

The logic gates commonly implemented in TTL/CMOS ICs are two-input OR, AND and XOR, their negations and NOT. I will interpret "IC" in the question as one of these logic gates, so we are looking to use at most three logic gates to represent the given expression. Let's simplify it on paper first: $$ABC+A'BC+AB'C+A'B'C'$$ $$=(AB+A'B+AB')C+(A+B)'C'$$ $$=(A+B)C+(A+B)'C'$$ $$=(A+B)\odot C$$ where $\odot$ is the XNOR gate, returning 1 iff both inputs are equal.

Immediately we have the admissible representations (A OR B) XNOR C and (if XNOR is not available as a single chip) NOT((A OR B) XOR C).