Consider two types of instructions for manipulating disjoint sets. FIND(x) computes the name of the (unique) set containing element x. UNION(A,B,C) combines sets A and B into a new set named C. We examinee a known algorithm for implementing sequences of these instructions. We show that if f(n) is the maximum time required by any sequence of n instructions, $k1n\alpha (n)\le f(n)\le k2n\log \ast (n)$ for some constants $k1$ and $k2$, where $\log \ast (n)=min\{i|\log i(n)\le 1\}$ and $\alpha (n)$ is a recursively defined function which satisfies $\alpha (n)\to \infty$ as $n\to \infty$. Thus the set union algorithm is $O(n\log \ast (n))$ but not $O(n)$. Keywords and phrases: algorithm, complexity, equivalence, partition, set union, tree.