function and relation

**novice** · May 3rd 2010, 01:09 PM

The examples of equivalence classes in my book are mostly related to integers. I would like to know this:

Can a continuous function have equivalence classes?

**Defunkt** · May 3rd 2010, 01:12 PM

What conditions need to be satisfied for a continuous function to be:

1) A well-defined function
2) An equivalence relation?

Answer those and you will get your result; note that you're not even considering the continuity.

**novice** · May 3rd 2010, 01:19 PM

Do you mind showing a simple example for someone like me?

**Defunkt** · May 3rd 2010, 01:32 PM

Sure:

Any function is a relation. Specifically, a function $f:A \to B$ is a relation $f \subseteq A \times B$ , such that for any $a \in A$ , there is exactly one $b \in B$ such that $(a,b) \in f$ .

Now, for f to be an equivalence relation, its domain has to be its range - $f: A \to A$ .

Specifically, it also has to be reflexive. That is, for each $a \in A, \ (a,a) \in f$ .

What does that tell you about f?

**novice** · May 3rd 2010, 01:40 PM

Very interesting. It looks like an identity function, but what does the equivalence classes look like since the points are so dense?

**novice** · May 3rd 2010, 01:42 PM

Oh, yeah, you already said about the continuity being irrelevant. Thank you for your valuable time.

**Defunkt** · May 3rd 2010, 01:47 PM

Well, what I meant was that you can get the result without assuming continuity, but the function you get (identity function) is always continuous. The equivalence class of $x \in \mathbb{R}$ is $[x] = \{ y \in \mathbb{R} : (x,y) \in f \}$ $= \{ y \in \mathbb{R} : f(x)=y \} = \{ y \in \mathbb{R} : x = y \} = \{ x \}$

So each point's equivalence class is itself.

**novice** · May 3rd 2010, 01:55 PM

Originally Posted by Defunkt

The equivalence class of $x \in \mathbb{R}$ is $[x] = \{ y \in \mathbb{R} : (x,y) \in f \}$ $= \{ y \in \mathbb{R} : f(x)=y \} = \{ y \in \mathbb{R} : x = y \} = \{ x \}$

So each point's equivalence class is itself.

Aha, that's what I was hoping to see.

Beautify! Thanks again.

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