Minimum excluded positive integer ($\min ({\mathbb{Z}}_{>0}\setminus S)$)

For a finite set $S$ of positive integers, its minimum excluded positive integer isFor example, $S=\{1,2,4\}$ gives three, while $S=\{2,4\}$ gives one. The answer lies between one and $|S|+1$, since those $|S|+1$ candidates cannot all lie in $S$.

The ordinary minimum excludant permits zero and would be zero for either example, because $S$ contains only positive integers. Choosing the positive or nonnegative convention is therefore part of the definition of the desired result.