Using Boolean identities, reduce the given Boolean expression: A(A+1) + A(B+0) + C.1

Boolean algebra provides identities that let an expression be simplified into its shortest equivalent form without changing what it represents logically.

Start with the given expression:

\[ A(A+1) + A(B+0) + C.1 \]

Simplify each term using standard Boolean identities:

• The identity \( X + 1 = 1 \) gives \( A + 1 = 1 \), so \( A(A+1) = A \cdot 1 = A \).
• The identity \( X + 0 = X \) gives \( B + 0 = B \), so \( A(B+0) = A \cdot B = A.B \).
• The identity \( X \cdot 1 = X \) gives \( C.1 = C \).

Substituting these simplified terms back gives:

\[ A + A.B + C \]

Next, apply the absorption law, which states that \( A + A.B = A \), since if \(A\) is already true the whole expression is true regardless of \(B\). This reduces the expression to:

\[ A + C \]

This matches the expression built from the sum of \(A\) and \(C\) alone. The other listed expressions either fail to apply the absorption law correctly or retain a term (such as \(B\)) that the simplification actually eliminates.

When simplifying Boolean expressions, always resolve identity laws (\(X+1=1\), \(X+0=X\), \(X.1=X\)) first, then check for absorption patterns like \(A + A.B\) before concluding the simplification is complete.