Let $X$ be the set $\{1,2,3,4\}$ and also that $$R = \{(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)\}$$

How do I how that $R$ is an equivalence relation; and also its equivalence classes?

I got equivalence class $\{1,2\}$, $\{3,4\}$ but I'm not sure if it's right.

Help appreciated!

  • $\begingroup$ You have to check the three properties explicitly: reflexive, symmetric, transitive. $\endgroup$ – Sammy Black Apr 5 '13 at 15:45
  • $\begingroup$ Does $R$ satisfy all three conditions for an equivalence relation as stated on Wikipedia? $\endgroup$ – Lord_Farin Apr 5 '13 at 15:45
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    $\begingroup$ Those can't be the equivalence classes because you have $(1,3)$ in your relation so $1$ and $3$ must be in the same class. $\endgroup$ – xavierm02 Apr 5 '13 at 15:46
  • $\begingroup$ It looks like this relation is the $\leq$ relation. $\endgroup$ – Pedro M. Apr 5 '13 at 15:47
  • $\begingroup$ @amWhy Ok, but I never said $\leq$ was an equivalence relation. $\endgroup$ – Pedro M. Apr 5 '13 at 15:57

Caution: the originally posted relation is not an equivalence relation.

Can you show that symmetry fails? For example: $(1, 3) \in R$, but $(3, 1) \notin R$

For your second problem, can you show that it satisfies reflexivity, symmetry, and transitivity?


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