1. The problem asks to identify Boyle's law in mathematical form. 2. Boyle's law states that for a given amount of gas at constant temperature, the pressure $P$ and volume $V$ of the gas are inversely proportional. 3. This means the product of pressure and volume is constant: $$P \times V = k$$ where $k$ is a constant. 4. Another common form relates two states of the gas: $$P_1 \times V_1 = P_2 \times V_2$$ showing that the product of pressure and volume before and after a change remains the same if temperature is constant. 5. Therefore, options (a) and (c) both correctly represent Boyle's law. 6. Option (b) $\frac{P}{V} = k$ is incorrect because pressure and volume are inversely proportional, not directly proportional. 7. The correct answer is (d) either (a) or (c).