If A={ 1, 2,3}, these five different types are divided into
{{ 1}, {2}, {3}}; {{ 1}, {2,3}}; {{ 1,3}, {2}}; {{ 1,2}, {3}}; {{ 1, 2, 3}};
The corresponding equivalent relationship 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 Bn different divisions (equivalence relations).
Bn=2n! /((n+ 1)n! n! ), such as a group of four elements, can determine 14 equivalence relations.