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Prove the Boolean identity: (A+B)(A+C) = A + BC

April 5, 2026Computer Science

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Answer

A + BC

4 steps

Here's the proof using Boolean theorems:

We need to prove: $(A+B)(A+C) = A + BC$

Start with the Left Hand Side (LHS) and apply the distributive law ( $X(Y+Z) = XY + XZ$ ).
$(A+B)(A+C) = A \cdot A + A \cdot C + B \cdot A + B \cdot C$
Apply the idempotent law ( $A \cdot A = A$ ) and the commutative law ( $B \cdot A = A \cdot B$ ).
$= A + AC + AB + BC$
Apply the absorption law ( $A + AC = A$ ).
$= (A + AC) + AB + BC$ $= A + AB + BC$
Apply the absorption law again ( $A + AB = A$ ).
$= (A + AB) + BC$ $= A + BC$
This is the Right Hand Side (RHS).

Thus, we have proven that $(A+B)(A+C) = A + BC$ .
$A + BC$

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