20.3.1 ${ }^{\star \star}$ Set $A$ consists of 100 non-negative integers, set $S$ consists of all numbers of the form $x+y$, where $x$, $y \in A$ (allowing $x=y$), how many numbers can $S$ have at most? At least?
Let $A=\left\{a_{1}, a_{2}, \cdots, a_{100}\right\}$, then $x+y$ must be one of the following forms:
$$
\begin{array}{c}
a_{1}+a_{1}, a_{1}+a_{2}, a_{1}+a_{3}, \cdots, a_{1}+a_{100} ; \\
a_{2}+a_{2}, a_{2}+a_{3}, \cdots, a_{2}+a_{100} ; \\
\cdots \\
a_{100}+a_{100} .
\end{array}
$$

Therefore, $|S| \leqslant 100+99+\cdots+1=5050$.
When $A=\left\{1,2,2^{2}, \cdots, 2^{99}\right\}$, $|S|=5050$. Let $a_{1}<a_{2}<\cdots<a_{100}$, then
$$
a_{1}+a_{1}, a_{1}+a_{2}, \cdots, a_{1}+a_{100}, a_{2}+a_{100}, a_{3}+a_{100}, \cdots, a_{100}+a_{100}
$$

are all distinct, so $|S| \geqslant 199$.
When $A=\{0,1,2, \cdots, 99\}$, $|S|=199$.
Therefore, $S$ can have at most 5050 numbers and at least 199 numbers.