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

The problem asks us to identify the correct logic circuit from the given options that corresponds to the provided truth table for inputs A and B, and output Y.

Concept Used:

To solve this problem, we first need to determine the Boolean expression for the output Y based on the inputs A and B from the truth table. Then, for each of the given logic circuits, we will derive its corresponding Boolean expression. The correct circuit will be the one whose expression matches the one derived from the truth table.

The basic logic gates and their Boolean operations are:

• OR gate: The output is 1 if at least one input is 1. Represented by \( + \). So, \( Y = A + B \).
• AND gate: The output is 1 only if all inputs are 1. Represented by \( \cdot \). So, \( Y = A \cdot B \).
• NOT gate (Inverter): The output is the inverse of the input. Represented by a bar over the variable. So, \( Y = \bar{A} \).

We will also use the absorption law of Boolean algebra: \( X \cdot (X + Z) = X \).

Step-by-Step Solution:

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

The truth table is:

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

By observing the table, we can see that the output Y is 0 whenever the input A is 0, and the output Y is 1 whenever the input A is 1. The value of Y is independent of the input B. Therefore, the Boolean expression for the output is simply:

\[ Y = A \]

Step 2: Analyze each of the given logic circuits to find their Boolean expressions.

Circuit (1):

The inputs A and B go into an OR gate, giving the output \( A + B \). This output and the input B are then fed into an AND gate. The final output Y is:

\[ Y = (A + B) \cdot B \]

Using the distributive law: \( Y = A \cdot B + B \cdot B = A \cdot B + B \). Using the absorption law, this simplifies to \( Y = B \). This does not match \( Y = A \).

Circuit (2):

The inputs A and B go into an OR gate, giving the output \( A + B \). This output and the input A are then fed into an AND gate. The final output Y is:

\[ Y = (A + B) \cdot A \]

Using the distributive law: \( Y = A \cdot A + B \cdot A \). Since \( A \cdot A = A \), this becomes \( Y = A + A \cdot B \). Factoring out A gives \( Y = A(1 + B) \). Since \( 1 + B = 1 \), the expression simplifies to:

\[ Y = A \]

This expression matches the expression derived from the truth table.

Circuit (3):

The inputs A and B go into an OR gate, giving \( A + B \). The input B is inverted to \( \bar{B} \). These two results are fed into an AND gate. The final output Y is:

\[ Y = (A + B) \cdot \bar{B} = A \cdot \bar{B} + B \cdot \bar{B} = A \cdot \bar{B} + 0 = A \cdot \bar{B} \]

This expression is 1 only when A=1 and B=0, which does not match the truth table.

Circuit (4):

The inputs A and B go into an OR gate, giving \( A + B \). The input A is inverted to \( \bar{A} \). These two results are fed into an AND gate. The final output Y is:

\[ Y = (A + B) \cdot \bar{A} = A \cdot \bar{A} + B \cdot \bar{A} = 0 + B \cdot \bar{A} = \bar{A} \cdot B \]

This expression is 1 only when A=0 and B=1, which does not match the truth table.

Final Computation & Result:

After analyzing all the circuits, only the circuit in option (2) produces the Boolean expression \( Y = A \), which is consistent with the given truth table. Therefore, the correct logic circuit is option (2).

The correct option is (2).

Answer 2

The truth table shows that the output Y is 1 only when either A is 1 and B is 0, or both A and B are 1. 
This can be represented as Y = (A AND NOT B) OR (A AND B).
Logic circuit (2) contains an AND gate, and an OR gate, and a NOT gate which matches to the truth table.
So the answer is (2).