Question

The value of \(Q\) in the given digital circuit is

• A. \[ Q=A+B+B.C(B+C) \]
• B. \[ Q=A.B+B.C(B+C) \]
• C. \[ Q=A+B+B.C(B.C) \]
• D. \[ Q=A.B+\overline{B}\,\overline{C}(B+C) \]

Hint:

Useful Boolean identities: \[ A+A.B=A, \] \[ A(A+B)=A, \] \[ (A+B)C=AC+BC. \] Always evaluate a digital circuit gate-by-gate from inputs to output.

Answer 1

The Correct Option is B

Concept: Identify the output of each logic gate and combine them step by step.

Step 1: Find the output of the top AND gate. Inputs are \(A\) and \(B\). Therefore, \[ X=A.B. \]

Step 2: Find the output of the middle OR gate. Inputs are \(B\) and \(C\). Hence, \[ Y=B+C. \]

Step 3: Find the output of the lower AND gate. Inputs are \(B\) and \(C\). Therefore, \[ Z=B.C. \]

Step 4: Find the output of the next AND gate. The outputs \(Y\) and \(Z\) are fed into an AND gate. Thus, \[ W=Y.Z. \] \[ W=(B+C)(B.C). \] \[ W=B.C(B+C). \]

Step 5: Find the final output. The final OR gate combines \(X\) and \(W\). Hence, \[ Q=X+W. \] \[ Q=A.B+B.C(B+C). \] Therefore, \[ \boxed{ Q=A.B+B.C(B+C) } \] \[ \boxed{\text{Answer = (B)}} \]