Question

For the given logic gate, find the output function.

• A. \( \bar{A}\cdot\bar{B} + C + D \)
• B. \( \bar{A} + \bar{B} + \bar{C}\cdot\bar{D} \)
• C. \( AB + CD \)
• D. \( AB + \bar{C}\cdot\bar{D} \)

Hint:

Useful identities:
\( (AB)' = A' + B' \)
\( (A + B)' = A'B' \) Always simplify step by step.

Answer 1

The Correct Option is D

Concept: To find the output of a logic circuit:
Write the Boolean expression gate by gate
Apply NOT operations carefully
Use De Morgan’s laws for simplification
Step 1: Analyze the upper branch. Inputs \(A\) and \(B\) go into an AND gate followed by a NOT gate. \[ \text{Upper output} = (A \cdot B)' \]
Step 2: Analyze the lower branch. Inputs \(C\) and \(D\) go into an OR gate. \[ \text{Lower output} = C + D \]
Step 3: Combine both branches. The outputs of the two branches go into an AND gate followed by a NOT gate. \[ Y = \big[(A \cdot B)' \cdot (C + D)\big]' \]
Step 4: Apply De Morgan’s theorem. \[ Y = (A \cdot B) + (C + D)' \] \[ (C + D)' = \bar{C}\cdot\bar{D} \] \[ Y = AB + \bar{C}\cdot\bar{D} \] \[ \boxed{Y = AB + \bar{C}\cdot\bar{D}} \]