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.