Y is the condenser*. Its function is to cool the hot vapor, causing it to condense back into liquid.

a)

• Y is the condenser. Its function is to cool the hot vapor, causing it to condense back into liquid.
• X is a thermometer. Its function is to measure the temperature of the vapor that is distilling.

The final answer is $\boxed{\text{Y: Condenser, X: Thermometer}}$

b) The method illustrated is distillation, which is used to separate miscible liquids with different boiling points.
i. Ethanol and water
ii. Methanol and water

The final answer is $\boxed{\text{i. Ethanol and water, ii. Methanol and water}}$

c) The property that makes it possible to separate the mixture using distillation is the difference in their boiling points.

The final answer is $\boxed{\text{Difference in boiling points}}$