This paper presents an analysis of the efficiency of a set union algorithm that combines two operations: FIND and UNION. The algorithm is used to manage disjoint sets of elements, where FIND(x) returns the set containing element x, and UNION(A, B, C) merges sets A and B into a new set C. The paper shows that the worst-case running time of this algorithm is O(m α(m, n)), where α(m, n) is a very slowly growing function related to the inverse of Ackermann's function. This bound is tight to within a constant factor. The algorithm uses two key techniques: collapsing and weighted union. Collapsing ensures that after a FIND operation, all nodes reached during the FIND are directly connected to the root of the tree, which speeds up subsequent FIND operations. The weighted union rule merges smaller sets into larger ones, which helps maintain a balanced tree structure. The paper provides both upper and lower bounds on the algorithm's performance. The upper bound is derived by analyzing the maximum time required for a sequence of FIND and UNION operations, considering the effects of collapsing and weighted union. The lower bound is established by constructing scenarios where the algorithm's performance is minimized, demonstrating that the upper bound is tight. The analysis shows that the algorithm's worst-case running time is significantly better than previously known algorithms, which have higher time complexity. The result is important because it provides a theoretical foundation for understanding the efficiency of set union algorithms, which are used in various applications such as finding minimum spanning trees, computing dominators in graphs, and handling equivalence relations. The paper concludes that the algorithm is optimal to within a constant factor and that further research is needed to determine if a linear-time algorithm exists for the online set union problem.This paper presents an analysis of the efficiency of a set union algorithm that combines two operations: FIND and UNION. The algorithm is used to manage disjoint sets of elements, where FIND(x) returns the set containing element x, and UNION(A, B, C) merges sets A and B into a new set C. The paper shows that the worst-case running time of this algorithm is O(m α(m, n)), where α(m, n) is a very slowly growing function related to the inverse of Ackermann's function. This bound is tight to within a constant factor. The algorithm uses two key techniques: collapsing and weighted union. Collapsing ensures that after a FIND operation, all nodes reached during the FIND are directly connected to the root of the tree, which speeds up subsequent FIND operations. The weighted union rule merges smaller sets into larger ones, which helps maintain a balanced tree structure. The paper provides both upper and lower bounds on the algorithm's performance. The upper bound is derived by analyzing the maximum time required for a sequence of FIND and UNION operations, considering the effects of collapsing and weighted union. The lower bound is established by constructing scenarios where the algorithm's performance is minimized, demonstrating that the upper bound is tight. The analysis shows that the algorithm's worst-case running time is significantly better than previously known algorithms, which have higher time complexity. The result is important because it provides a theoretical foundation for understanding the efficiency of set union algorithms, which are used in various applications such as finding minimum spanning trees, computing dominators in graphs, and handling equivalence relations. The paper concludes that the algorithm is optimal to within a constant factor and that further research is needed to determine if a linear-time algorithm exists for the online set union problem.