1. Given the set $S=\{1,2,3, \cdots, 2000,2001\}$, where a subset $T$ has the property that for any three elements $x, y, z$ in $T$, $x+y \neq z$, what is the maximum number of elements in $T$? Maximum value
1. Solve 1001.

Let $T$ contain $k$ elements. Subtracting each of the other elements from the largest element, the differences obtained are not in $T$, so $k + (k-1) \leqslant 2001$, which means $k \leqslant 1001$. The subset $\{1001,1002, \cdots, 2001\}$ of $S$ contains 1001 elements, which meets the requirement, so $k_{\max }=1001$.