1. ## Equivalence Relation

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.*

Thanks.

2. Originally Posted by DeFacto
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.*

Thanks.
yes you got all the relations there. not sure exactly what you mean in the second part, but given partition on a set the relation "x is related to y, if they belong to the same partition" is an equivalence relation.

Bobak