A quick review of a family of disjoint sets: These can be construed as equivalence classes, sets of mutually related elements, where the relation is reflexive, symmetric and transitive (attributes of an equivalence relation.) Equivalence classes partition the universe: any element belongs to exactly one of them and no two intersect. (Making a "forward reference," we invoke vertices of a combinatorial graph spanned by a forest of trees; belonging to the same tree is an example of such an equivalence relation.)
To appreciate the relative sophistication of the problem, one can attempt to design implementation of ADT Partition with an eye on the complexity of its operations. The concept of the characteristic function of a set (boolean function returning True for a member of the set) may give a guidance: represent each set by a boolean array cooresponding to its characteristic function. This has several obvious disadvantages: each set needs an array of the size of the universe of the elements. To improve this, we replace the boolean arrays by a single array (indexed by elements) containing the names of the sets: for each element x the set Find(x). This implements Find in O(1) time, but the complexity of Union becomes O(n). Alternatively, each set could be represented by a list of its elements (linked, so that we can put them together quickly.) List headers can be stored in an array indexed by set names; in this implementation Union will be O(1), but the complexity of Find deteriorates to O(n).
To achieve a balance in the operations' complexities, we could employ a variety of tools in our toolbox; for instance, link elements of each set in the set name array above (to avoid accessing all the elements) and establish a translation table for set names (to be able to change only the name of a smaller set in a Union.) This leads to a logarithmic complexity of Union, but even with different organization of data (eg., balanced tree of set elements instead of a list), \Omega(\log n) seems to be the lower bound for at least one of the operations.