3.1.10 problem 12

(a)

https://drive.google.com/file/d/1vAAxLU7hvihAHOEcHx8Nc-9xapGlzc-I/view?usp=sharing

(b)

Let $I\subset X$ be the subset of the support where $P_1\left(x\right)<P_2\left(x\right)$. Then $$\sum _{x\in X}{P}_1\left(x\right)=\sum _{x\in I}{P}_1\left(x\right)+\sum _{x\in X\setminus I}{P}_1\left(x\right)<\sum _{x\in I}{P}_2\left(x\right)+\sum _{x\in X\setminus I}{P}_2\left(x\right)=1.$$

Thus, having such a property in PMFs is impossible.