Question

List all equivalence relations on {0, 1, 2, 3}.

Answer 1

A Relation 'R' on a set 'A' is said to be equivalence relation on 'A' if 'R' is

1)Reflexive relation

2)Symmetric relation

3)Transitive relation

Now Considering A={0,1,2,3}{0,1,2,3}

We get (0,0),(0,2),(2,0),(2,2),(2,3),(3,2),(3,3)(0,0),(0,2),(2,0),(2,2),(2,3),(3,2),(3,3)

in fact it is not equivalence relation since it is not reflexive.

The smallest equivalence relation that can be formed is{(0,0),(1,1),(2,2),(3,3)}.{(0,0),(1,1),(2,2),(3,3)}.

The largest equivalence relation isA×A.

The number of equivalence relation can be found using Bell number  , for n=4 ,B4=15.