2. Let $S$ be a subset of $\{1,2, \cdots, 9\}$ such that the sum of any two distinct elements of $S$ is unique. How many elements can $S$ have at most?
(2002, Canadian Mathematical Olympiad)
(When $S=\{1,2,3,5,8\}$, $S$ meets the requirements of the problem. If $T \subseteq\{1,2, \cdots, 9\},|T| \geqslant 6$, then since the sum of any two different numbers in $T$ is between 3 and 17, at most 15 different sum numbers can be formed. And choosing any two numbers from $T$, there are at least $\mathrm{C}_{6}^{2}=15$ ways. If $T$ meets the condition, then the sum numbers must include both 3 and 17, meaning that $1,2,8,9$ must all appear in $T$, but in this case $1+9=2+8$, making $T$ not meet the requirement. Therefore, $S$ can have at most 5 elements.)