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

The calculator will simplify the boolean expression (A⋅B)+(B⋅C)+(A⋅C‾)\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 (A⋅B)+(B⋅C)+(A⋅C‾)\left(A \cdot B\right) + \left(B \cdot C\right) + \left(A \cdot \overline{C}\right).

Solution

Apply the consensus law (x⋅y)+(x‾⋅z)+(y⋅z)=(x⋅y)+(x‾⋅z)\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=Cx = C, y=By = B, and z=Az = A:

((A⋅B)+(B⋅C)+(A⋅C‾))=((C⋅B)+(C‾⋅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

(A⋅B)+(B⋅C)+(A⋅C‾)=(C⋅B)+(C‾⋅A)\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)