Question

Choose the correct logic circuit for the given truth table having inputs A and B.

• A.
• B.
• C.
• D.

Hint:

Analyze each logic gate combination and compare its output with the given truth table to find the correct match.

Answer 1

The Correct Option is B

Logic Circuit and Boolean Expressions

Concept Used:

To address this, we first derive the Boolean expression for output Y from the given truth table. Subsequently, we determine the Boolean expression for each of the provided logic circuits. The correct circuit is the one whose derived Boolean expression matches that obtained from the truth table.

Logic Gates and Their Boolean Operations:

• OR gate: Output is 1 if at least one input is 1. Boolean expression: \( Y = A + B \).
• AND gate: Output is 1 only if all inputs are 1. Boolean expression: \( Y = A \cdot B \).
• NOT gate: Output is the inverse of the input. Boolean expression: \( Y = \bar{A} \).

Step-by-Step Solution:

Step 1: Analyze the truth table to determine the Boolean expression for Y.

The truth table is:

A	B	Y
0	0	0
0	1	0
1	0	1
1	1	1

Observation of the truth table indicates that Y is 0 when A is 0, and Y is 1 when A is 1. The value of Y is unaffected by the input B. Consequently, the Boolean expression for the output is:

\( Y = A \)

Step 2: Evaluate the Boolean expressions for each of the provided logic circuits.

Circuit (1):

The output expression for this circuit is \( Y = (A + B) \cdot B \). Simplifying:

\( Y = A \cdot B + B \), which simplifies to \( Y = B \) by absorption law. This does not match \( Y = A \).

Circuit (2):

The output expression for this circuit is \( Y = (A + B) \cdot A \). Simplifying:

\( Y = A \cdot A + A \cdot B = A + A \cdot B = A(1 + B) = A \). This matches the Boolean expression derived from the truth table.

Circuit (3):

The output expression for this circuit is \( Y = (A + B) \cdot \bar{B} = A \cdot \bar{B} + 0 = A \cdot \bar{B} \), which only holds when A=1 and B=0, which is inconsistent with the truth table.

Circuit (4):

The output expression for this circuit is \( Y = (A + B) \cdot \bar{A} = \bar{A} \cdot B \), which only holds when A=0 and B=1, which is inconsistent with the truth table.

Final Result:

Upon analyzing all circuits, only circuit (2) yields the Boolean expression \( Y = A \), which matches the truth table. Therefore, logic circuit (2) is the correct solution.

The correct option is (2).