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

The output $$Y$$ is directly dependent only on input $$A$$. The Boolean expression is simply $$Y = A$$

Option A:
$$ Y = (A + B) \cdot B $$

$$ Y = A \cdot B + B \cdot B $$

Since $$B \cdot B = B$$:

$$ Y = A \cdot B + B $$

$$ Y = B \cdot (A + 1) $$

Since $$A + 1 = 1$$:

$$ Y = B \cdot 1 $$

Option B:
$$ Y = A \cdot (A + B) $$
$$ Y = A \cdot A + A \cdot B $$
$$ Y = A + A \cdot B $$
$$ Y = A \cdot (1 + B) $$
$$ Y = A \cdot 1 $$
$$ Y = A $$

Option C:
$$ Y = A \cdot (\overline{A + B}) $$
$$ Y = A \cdot (\overline{A} \cdot \overline{B}) $$
$$ Y = (A \cdot \overline{A}) \cdot \overline{B} $$
Since $$A \cdot \overline{A} = 0$$:
$$ Y = 0 \cdot \overline{B} $$
$$ Y = 0 $$

Option D:
$$ Y = \overline{A} \cdot (A + B) $$
$$ Y = \overline{A} \cdot A + \overline{A} \cdot B $$
$$ Y = 0 + \overline{A} \cdot B $$
$$ Y = \overline{A} \cdot B $$