# Index Set

Let be a set. A set is called an “index set” of the set if and that set are equipollent. I.e., a set is called an index set of if there exists a bijective function (injective and surjective) between and . As is clear from the definition, the number of all the index sets of a set may be more than one, even infinite. Besides, we can say from the definition: there exists at least one index set of any set. Because, the function is bijective. This is a trivial example since the set is indexed by itself. If we indexed every set with itself, the indexing operation would be meaningless. We must give a reinforcement example: Let be . We will give three index sets for this set: The sets , and can be given as index sets of . (Can be given more index sets for this set). The type of index set that is widely used by the mathematicians is since it one by one counts the elements of . In general, can be chosen as an index set for a set with elements. An index set of a set is directly associated with the cardinality of that set. Since the cardinality relation on any family of sets is an equivalence relation, an index set of a set can be actually considered as the most reasonable “representation of class” of the equivalence class of a set. For example, the most reasonable index set of any countable set is naturally the set of the natural numbers. , or is widely used as an index set for a set that is equipollent with the set of the real numbers.