# Equivalence classes

I'm posting this question and answers to see if I am on the right track here, just want to be sure I understand or don't understand.

Bellow I will list some equivalence relations over the set $S= \{1,2,3,4\}$ the assignment is to find the equivalence classes to $[1]$

$\{<1,1>,<2,2>,<3,3>,<4,4>\}, [1] = \{1\}$

$\{<1,1>,<2,2>,<3,3>,<4,4>,<1,2>,<2,1>\}, [1] = \{1,2\}$

$\{<1,1>,<2,2>,<3,3>,<4,4>,<1,3>,<3,1>\}, [1] = \{1,3\}$

$\{<1,1>,<2,2>,<3,3>,<4,4>,<1,4>,<4,1>,<2,3>,<3,2>\}, [1] = \{1,4\}$

$\{<1,1>,<2,2>,<3,3>,<4,4>,<2,3>,<3,2>\}, [1] = \{1\}$

$\{<1,1>,<2,2>,<3,3>,<4,4>,<1,2>,<1,4>,<2,1>,<2,4>,<4,1>,<4,2>\}, [1] = \{1,2,4\}$

So if my answers are correct then great, if not what do I need to look at?

Cheers

• Draw some pictures with the elements as nodes and relations as edges. Find the connected component of $1$. – lhf Nov 14 '13 at 13:01

It is right except the first relation: it is not an equivalence (maybe you forgot to write $<4,4>$).
• Yeah I did forget that, could you tell me a little more indepth as to why it is wrong with $<4,4>$ missing? I just want to make sure I got this locked down – user2008560 Nov 14 '13 at 13:10
• Every equivalence relation $R$ is reflexive, i.e. $\forall x \ <x,x>\in R$. – Boris Novikov Nov 14 '13 at 13:14