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A relation R on the set $M = \{1,2,3,4,5\}$ is defined by

$R=\{(1,1),(1,2),(2,1),(2,2),(2,4),(3,3),(3,5),(4,2),(4,4),(5,3),(5,5)\}$.

What are the equivalence classes of R?


I know what equivalence classes are but I'm really confused as to how to apply them to this question. Would someone be able to nudege me in the right direction? Are all the elements of R of the form $(a,a)$ where $a\in M$ in one equivalence class?

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  • $\begingroup$ "I know what equivalence classes are" then what are they? $\endgroup$ – Kenny Lau Sep 4 '17 at 10:27
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    $\begingroup$ It doesn't make sense because $R$ is not an equivalent relation: $(1,2) \in R$ and $(2,4) \in R$, but $(1,4) \notin R$. $\endgroup$ – Kenny Lau Sep 4 '17 at 10:27
  • $\begingroup$ That's true, I didn't form the relation correctly. No wonder I was confused, thanks for the heads up :) $\endgroup$ – Sonjov Sep 4 '17 at 10:29
  • $\begingroup$ Although both $(1, 2)$ and $(2, 4)$ are in $R$, $(1, 4)$ is not in $R$, meaning that $R$ is not transitive and hence not an equivalence relation. $\endgroup$ – Saaqib Mahmood Sep 4 '17 at 10:33
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    $\begingroup$ Equivalence class represented e.g by $1$ is the set $[1]=\{x\in M\mid (1,x)\in R\}$. $\endgroup$ – drhab Sep 4 '17 at 10:42

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