
Among the given compounds, compound (d) is aromatic, which greatly enhances its stability due to resonance stabilization. Compound (a) is antiaromatic, making it the least stable among the given options.
Compound(b) and compound (c) have sp3 hybridized carbon atoms bearing the negative charge. Based on angle strain theory, compound (c) is more stable than compound (b) due to its ring structure.

To obtain the given truth table, the following logic gate should be placed at G:
Let a line passing through the point $ (4,1,0) $ intersect the line $ L_1: \frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} $ at the point $ A(\alpha, \beta, \gamma) $ and the line $ L_2: x - 6 = y = -z + 4 $ at the point $ B(a, b, c) $. Then $ \begin{vmatrix} 1 & 0 & 1 \\ \alpha & \beta & \gamma \\ a & b & c \end{vmatrix} \text{ is equal to} $