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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.

FERN4V3
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1 Answers1

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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.

PTrivedi
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  • So in general, we can add a product of all variables involved to the equation without altering it ? and then use it for simplifying ? – FERN4V3 Apr 05 '22 at 02:32
  • So $ABC$ was already in the product so we can add it to the product as many times without altering it. For simplicity consider $D = D + D$. The reason this is true is because if $D = T$ then we have $T = T + T$ and if $D = F$ then we can $F = F + F$. So it does not alter the sum. Then by induction we can add as many times as we'd like. – PTrivedi Apr 05 '22 at 02:33