Each equivalence relation on a set corresponds to a partition of the set, and each partition of the set corresponds to an equivalence relation of the set. Different equivalence relations correspond to different divisions of sets, so there are many different equivalence relations. The ternary set * * * has five different divisions (there are 1 block and 1 block, and there are three blocks).

If A={ 1, 2,3}, the five differences are divided into:

{{ 1},{2},{3}}; {{ 1},{2,3}}; {{ 1,3},{2}}; {{ 1,2},{3}}; { 1, 2, 3}};

The corresponding equivalence relation is:

R 1={( 1， 1)，(2，2)，(3，3)}；

R2={( 1， 1)，(2，2)，(2，3)，(3，2)，(3，3)}；

R3={( 1， 1)，( 1，3)，(3， 1)，(2，2)，(3，3)}；

R4={( 1， 1)，( 1，2)，(2 1)，(2，2)，(3，3)}；

R5={( 1， 1)、(2，2)、(3，3)、( 1，2)、(2，3)、(3，2)、( 1，3)、(3， 1)}；

Generally speaking, the set of n elements has different divisions (equivalence relations) of Bn, Bn is called Catalan number, Bn=2n! /((n+ 1)n！ n！ ), such as a group of four elements, can determine 14 equivalence relations.