Question

Which of the following logic expressions represents the logic diagram shown?

• A. X=AB'+A'B
• B. X=(AB)'+AB
• C. X=(AB)'+A'B'
• D. X=A'B'+AB

Hint:

Logic Circuit Analysis. Trace the signals through each gate. Use Boolean identities and De Morgan's theorems (\(\overline{A+B=\overline{A\overline{B\), \(\overline{AB=\overline{A+\overline{B\)) to simplify. Recognize standard functions: XOR=\(A\overline{B+\overline{AB\), XNOR=\(AB+\overline{A\overline{B\).

Answer 1

The Correct Option is D

Analyzing the logic diagram step-by-step: - The top AND gate receives inputs \(\overline{A}\) (from the NOT gate connected to A) and \(\overline{B}\) (from the NOT gate connected to B)
Its output is \( \overline{A} \cdot \overline{B} \)
- The bottom AND gate receives inputs A and B
Its output is \( A \cdot B \)
- The final gate is an XOR gate
Its output X is the XOR of the outputs from the two AND gates: $$ X = (\overline{A} \cdot \overline{B}) \oplus (A \cdot B) $$ - Let's evaluate the XOR operation: \( P \oplus Q = P\overline{Q} + \overline{P}Q \)
Here, \(P = \overline{A}\overline{B}\) and \(Q = AB\)
\(\overline{Q} = \overline{AB} = \overline{A} + \overline{B}\) (De Morgan's) \(\overline{P} = \overline{\overline{A}\overline{B}} = A + B\) (De Morgan's) $$ X = (\overline{A}\overline{B})(\overline{A} + \overline{B}) + (A + B)(AB) $$ $$ X = (\overline{A}\overline{B}\overline{A} + \overline{A}\overline{B}\overline{B}) + (AAB + BAB) $$ $$ X = (\overline{A}\overline{B} + \overline{A}\overline{B}) + (AB + AB) $$ $$ X = \overline{A}\overline{B} + AB $$ This resulting expression \(X = \overline{A}\overline{B} + AB\) is the definition of the XNOR function
This matches option (4)
(Note: The symbol for the final gate in the diagram is XOR, but the resulting function matching the correct option is XNOR
This implies either the symbol is drawn incorrectly or it's meant to be interpreted as XNOR in this context)