Which one of the following will be the output of the given circuit?

We need to determine the equivalent logic function for the given digital circuit combinations containing an AND gate, an OR gate, a NOT gate, and a final combination gate.

1. Analyze the Upper Path

• The two inputs $$A$$ and $$B$$ are fed into the first gate, which is a standard AND gate.

  Output of the AND gate = $$A \cdot B$$

• This output is immediately passed through an inverter, which is a NOT gate.

  Output after the inverter = $$\overline{A \cdot B}$$

2. Analyze the Lower Path

• The two inputs $$A$$ and $$B$$ are simultaneously fed into the lower gate, which is a standard OR gate.

  Output of the OR gate = $$A + B$$

3. Combine at the Final Gate

The outputs from both the upper path ($$\overline{A \cdot B}$$) and the lower path ($$A + B$$) serve as the inputs to the final AND gate to produce the final system output $$Y$$:

$$Y = (\overline{A \cdot B}) \cdot (A + B)$$

4. Simplify using Boolean Algebra

According to De Morgan's Law, the term $$\overline{A \cdot B}$$ can be expanded as $$\bar{A} + \bar{B}$$. Substituting this back into the expression for $$Y$$ gives:

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

Distributing the terms across the brackets:

$$Y = \bar{A}A + \bar{A}B + \bar{B}A + \bar{B}B$$

Since the complement laws state that any variable ANDed with its inverse equals zero ($$\bar{A}A = 0$$ and $$\bar{B}B = 0$$), the equation simplifies directly to:

$$Y = 0 + \bar{A}B + A\bar{B} + 0$$

$$Y = A\bar{B} + \bar{A}B$$

The resulting boolean expression ($$A\bar{B} + \bar{A}B$$) is the standard definition of an exclusive-OR operation.

Conclusion

The logic circuit behaves exactly like an XOR Gate, which corresponds to Option D.