Simplify: A̅B̅C + BC + AC

CONCEPT

Boolean algebra uses the complement law X + X̅ = 1, the identity law X·1 = X, and the absorption form X + X̅Y = X + Y.

A common factor may be taken outside a sum of product terms by the distributive law.

APPLICATION

1. Let F = A̅B̅C + BC + AC. Every term contains C, so F = C(A̅B̅ + B + A).

2. Apply X + X̅Y = X + Y with X = B and Y = A̅: B + B̅A̅ = B + A̅. Thus F = C(A + B + A̅).

3. Since A + A̅ = 1, the bracket is 1 + B = 1. Therefore F = C·1 = C.

CROSS-CHECK

If C = 0, all three original product terms are 0.

If C = 1, then A̅B̅ + B + A equals 1: when A or B is 1, A + B is 1; when A = B = 0, A̅B̅ is 1. Hence the original expression has exactly the same value as C in every case.

RESULT

The simplified expression is C.