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Consider the set $[3]$ and the equivalence relation defined by the graph: $$\{(1,1), (1,2), (2,1), (2,2), (3,3)\}.$$ I know this is an equivalence relation because it is symmetric, transitive, and reflexive. My professor says $1\sim 2$ since $\sim$ is reflexive, symmetric and transitive, but $1$ is not equivalent to $3$? Why is this the case?

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    $\begingroup$ $(1,3)$ is not one of the related pairs you gave us. Just to be clear $1\sim 2$ because $(1,2)$ is one of the pairs you gave us, not for the reasons you gave. $\endgroup$ – lulu Mar 15 '20 at 21:09
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    $\begingroup$ $1\sim2$ because $(1,2)\in R$ but $1\not\sim3$ because $(1,3)\not\in R$ $\endgroup$ – J. W. Tanner Mar 15 '20 at 21:10
  • $\begingroup$ Are your sure that your professor really said "1~ 2 since ~ is reflexive, symmetric and transitive"? That is why it is an equivalence relation but not why "1~ 2". 1~ 2 because (1, 2) is in that set, 1≁3 because (1, 3) is not in the set. $\endgroup$ – user247327 Jun 26 '20 at 19:58
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If $1$ were equivalent to $3$, then $(1,3)$ would be in $R$. Since $(1,3) \not \in R$, $1 \not \sim 3$. The equivalence of elements when you know all ordered pairs $(x,y) \in R$ is not going to be something you attempt to draw out from the properties of equivalence relations; it'll be quite clear since you'll know outright whether they're related or not.


Mostly posting this so this question can be finally considered to have an answer and thus be removed from the unanswered queue. Made it Community Wiki since I have nothing much of substance to add.

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