Hi, i'm studying some basic group theory at the moment, and i'm not sure whether or not i've got the jist of these things yet?
Question: Consider the set {1,2,3}, how many equivalence relations are there on this set? Justify your answer.
I've found what i think are 5:
{(1,1),(2,2),(3,3)}
{(1,1),(1,2),(2,1),(2,2),(3,3)}
{(1,1),(1,3),(3,1),(2,2),(3,3)}
{(1,1),(2,3),(3,2),(2,2),(3,3)}
{(1,1),(1,2),(2,1),(1,3),(3,1),(2,3),(3,2),(2,2),( 3,3)}
as these are the only ones that satisfy the transitivity, reflexive and symmetric properties. Is this correct?
Also,
*Given a partition {A_i : i belongs to I} of S describe, without proof, the unique equivalence relation on S whose equivalence classes form this partition.*
I'm not sure how to go about this. Any help would be greatly appreciated.
Thanks.