A disjoint set, also known as a union-find data structure, is used to track a collection of disjoint (non-overlapping) sets. This data structure supports two main operations: `find`, which determines the set to which a particular element belongs, and `union`, which merges two sets into one. Disjoint sets are particularly useful in scenarios where you need to keep track of connected components, such as in graph algorithms like Kruskal’s Minimum Spanning Tree (MST) algorithm. In Kruskal’s algorithm, the disjoint set is used to check whether adding an edge would form a cycle by determining if the two vertices connected by the edge belong to the same set. If they do, adding the edge would create a cycle, and it’s discarded; otherwise, the edge is added, and the two sets are merged. Disjoint sets are typically implemented using two techniques to optimize performance: union by rank and path compression. Union by rank ensures that the smaller set is always added to the larger set during a union operation, while path compression flattens the structure of the tree whenever `find` is called, making future operations faster. These optimizations ensure that both `find` and `union` operations run in nearly constant time, making disjoint sets an efficient solution for problems involving connected components, network connectivity, and clustering.