Simplify $$$\left(A \cdot B\right) + \left(B \cdot C\right) + \left(A \cdot \overline{C}\right)$$$

The calculator will simplify the boolean expression $$$\left(A \cdot B\right) + \left(B \cdot C\right) + \left(A \cdot \overline{C}\right)$$$, with steps shown.

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Your Input

Simplify the boolean expression $$$\left(A \cdot B\right) + \left(B \cdot C\right) + \left(A \cdot \overline{C}\right)$$$.

Solution

Apply the consensus law $$$\left(X \cdot Y\right) + \left(\overline{X} \cdot Z\right) + \left(Y \cdot Z\right) = \left(X \cdot Y\right) + \left(\overline{X} \cdot Z\right)$$$ with $$$X = C$$$, $$$Y = B$$$, and $$$Z = A$$$:

$${\color{red}\left(\left(A \cdot B\right) + \left(B \cdot C\right) + \left(A \cdot \overline{C}\right)\right)} = {\color{red}\left(\left(C \cdot B\right) + \left(\overline{C} \cdot A\right)\right)}$$

Answer

$$$\left(A \cdot B\right) + \left(B \cdot C\right) + \left(A \cdot \overline{C}\right) = \left(C \cdot B\right) + \left(\overline{C} \cdot A\right)$$$