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