# Why is {1, 2, 3} an equivalence relation?

I know the equality relation = is an equivalence relation in the set of real numbers. Because it can satisfy all three conditions of equivalence relation. (1) x=x, reflexive (2) x=y ⇒ y=x symmetric, (3) x=y∧y=z ⇒x=z, transitive

But what does {1, 2, 3} have have to do with equality relation (=), which is an equivalence relation?

• It is kind of an arbitrary remark. I think its purpose is to show that you can define equivalence relations on any set (you don't have to take $\mathbb{R}$, $\mathbb{N}$ or $\mathbb{Q}$, but you can take a finite set you define yourseld). Commented Dec 27, 2015 at 15:14
• I think you're just reading it incorrectly. The set $\{1,2,3\}$ is meant to be an example of a set of numbers rather than an equality relation. Commented Dec 27, 2015 at 15:21

## 2 Answers

You are simply parsing the English sentence incorrectly. It is

[The equality relation (=)] on [a set of numbers such as {1, 2, 3}] is [an equivalence relation].

not

[The equality relation (=) on a set of numbers] [such as {1, 2, 3}] is [an equivalence relation].

• +1 for being able to precisely trace the source of the OP's misunderstanding and clarifying it very efficiently. Commented Dec 27, 2015 at 18:13
• Let's all just speak Lojban so things like this don't ever happen. Commented Dec 28, 2015 at 0:43

The set $\{1,2,3\}$ is merely an example where it is easy to check that the properties of an equivalence relation hold. The equality relation is an equivalence relation on any set, for exactly the reasons you gave.