# Index Set

Let $X$ be a set. A set is called an “index set” of the set $X$ if $X$ and that set are equipollent. I.e., a set $I$ is called an index set of $X$ if there exists a bijective function (injective and surjective) between $I$ and $X$. 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 $I_{X}:X\to{X}$ is bijective. This is a trivial example since the set $X$ 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 $X=\{\diamondsuit, \heartsuit, \clubsuit, \spadesuit\}$. We will give three index sets for this set: The sets $I_1=X$, $I_2=\{1,2,3,4\}$ and $I_3=\{a,b,c,d\}$ can be given as index sets of $X$. (Can be given more index sets for this set). The type of index set that is widely used by the mathematicians is $I_{2}$ since it one by one counts the elements of $X$. In general, $I=\{1,2,\dots,n\}$ can be chosen as an index set for a set with $n$ 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. $\mathbb{R}$, $[0,1]$ or $(0,1)$ is widely used as an index set for a set that is equipollent with the set of the real numbers.