how pressure is inversely proportional to volume . explain with brief theory and proof .

Dear Student,

Let we take an example of balloon for the discussion of pressure and volume realtionship, if we press balloon by hand the volume decreases but the pressure gets increased by pressing continuously results in bursting of balloon by crossing the threshold of pressure. Vice versa when the balloon is blowed the moles of air "n" term in gas law increases leads to increase in volume hence the mathematical relationship as per Boyle's law is,

$$\begin{gathered} PV=k=nRT \\ PV=k \\ P \alpha \frac{1}{V} \to Boyles law \end{gathered}$$

Regards.