Basic Set Theory problem
Characteristic vector is a string representation of a set in this case.
For a set $S$, the characteristic vector is denoted by $X_S$. $X_\emptyset$ and $X_U$ are referring to the characteristic vector of $U$ (Universal set) and $\emptyset$ (Empty set) respectively.
In your example,
$$\text{If }n = 4 \text{ and } S = \{1, 3\} \text{, then } X_s = 1010.$$
We simply need to replace $S$ with a set $\{0, 1, 2, \dots n-1\}$ to find $X_U$ and with $\{\} = \emptyset$ to find $X_\emptyset$. I think you can leave $n$ as a variable unless it is fixed to some value.