Find the truth table for the function Y of A and B represented in the following figure.

We need to determine the truth table for the logic circuit function $$Y$$ shown in the diagram.

1. Analyze the Logic Gates

From the circuit diagram , the system consists of three distinct logic gates connected as follows:

• Top Gate (AND Gate):
  The inputs to this gate are $$A$$ and $$B$$. Its output expression is:

  $$Y_1 = A \cdot B$$

• Bottom Gate (NOT Gate):
  The input to this inverter is $$B$$. Its output expression is:

  $$Y_2 = \bar{B}$$

• Output Gate (OR Gate):
  This gate combines the intermediate outputs $$Y_1$$ and $$Y_2$$. The final boolean output expression for $$Y$$ is:

  $$Y = Y_1 + Y_2 = (A \cdot B) + \bar{B}$$

2. Construct the Truth Table Step-by-Step

Let's evaluate the output $$Y$$ for all four possible combinations of binary inputs ($$A$$ and $$B$$):

1. When $$A = 0$$ and $$B = 0$$:
   • $$A \cdot B = 0 \cdot 0 = 0$$
   • $$\bar{B} = \bar{0} = 1$$
   • $$Y = 0 + 1 = 1$$


2. When $$A = 0$$ and $$B = 1$$:
   • $$A \cdot B = 0 \cdot 1 = 0$$
   • $$\bar{B} = \bar{1} = 0$$
   • $$Y = 0 + 0 = 0$$


3. When $$A = 1$$ and $$B = 0$$:
   • $$A \cdot B = 1 \cdot 0 = 0$$
   • $$\bar{B} = \bar{0} = 1$$
   • $$Y = 0 + 1 = 1$$


4. When $$A = 1$$ and $$B = 1$$:
   • $$A \cdot B = 1 \cdot 1 = 1$$
   • $$\bar{B} = \bar{1} = 0$$
   • $$Y = 1 + 0 = 1$$

3. Compiled Truth Table

Summarizing our calculations into a structured table:

Conclusion

The calculated truth table matches the second choice(B).