0
$\begingroup$

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

$\endgroup$
  • $\begingroup$ Draw some pictures with the elements as nodes and relations as edges. Find the connected component of $1$. $\endgroup$ – lhf Nov 14 '13 at 13:01
0
$\begingroup$

It is right except the first relation: it is not an equivalence (maybe you forgot to write $<4,4>$).

$\endgroup$
  • $\begingroup$ 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 $\endgroup$ – user2008560 Nov 14 '13 at 13:10
  • 1
    $\begingroup$ Every equivalence relation $R$ is reflexive, i.e. $\forall x \ <x,x>\in R$. $\endgroup$ – Boris Novikov Nov 14 '13 at 13:14
  • $\begingroup$ Yes of course I see it now, I forgot that 4 was in the set S. Thank you for clearing it up! $\endgroup$ – user2008560 Nov 14 '13 at 13:16

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.