Question

Write the simplified form of the Boolean expression \( (A + C)(AD + AD') + AC + C \):

• A. A + C'
• B. A' + C
• C. A + D
• D. A + C

Hint:

Always look for simplification opportunities inside parentheses first. Key identities to look for are the complementation laws (\( A\bar{A}=0, A+\bar{A}=1 \)) and the absorption laws (\( A+AB=A, A(A+B)=A \)). They often lead to the quickest simplifications.

Answer 1

The Correct Option is D

Let's simplify the given Boolean expression step-by-step using the laws of Boolean algebra.
Expression: \( (A + C)(AD + A\bar{D}) + AC + C \)
 

Step 1: Simplify the innermost parenthesis. Using the distributive law, we can factor out A:
\[ AD + A\bar{D} = A(D + \bar{D}) \] By the law of complementation, \( D + \bar{D} = 1 \). So, \( A(1) = A \).

Step 2: Substitute the simplified part back into the expression. The expression now becomes:
\[ (A + C)(A) + AC + C \]

Step 3: Apply the distributive and absorption laws. \[ A(A + C) = AA + AC = A + AC \] (since \( AA = A \)) By the absorption law, \( A + AC = A \).
So the expression simplifies to:
\[ A + AC + C \]

Step 4: Apply the absorption law again. The expression is \( (A + AC) + C \). We know \( A + AC = A \). So we have \( A + C \). 

Alternatively, \( A + (AC + C) \). We know \( AC+C = C(A+1) = C \). So we have \( A + C \). The final simplified form is \( A + C \).