# Equivalence relations on the set [3].

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?

• $(1,3)$ 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. – lulu Mar 15 '20 at 21:09
• $1\sim2$ because $(1,2)\in R$ but $1\not\sim3$ because $(1,3)\not\in R$ – J. W. Tanner Mar 15 '20 at 21:10
• 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. – user247327 Jun 26 '20 at 19:58

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.