# Thread: Bijective Equivalence relation proof

1. ## Bijective Equivalence relation proof

Let U be a family of all finite sets. Define the relation ~1 on U as follows

A1 ~1 A2 iff there is a bijection f: A1->A2

Prove that ~1 is an equivalence relation and describe the equivalence classes.

I can't seem to remember how to do this. I forgot this part of Set Theory. Thank you to anyone for help!

2. Originally Posted by selkam47
Let U be a family of all finite sets. Define the relation ~1 on U as follows
A1 ~1 A2 iff there is a bijection f: A1->A2
Prove that ~1 is an equivalence relation and describe the equivalence classes. I can't seem to remember how to do this. I forgot this part of Set Theory.
Have you looked it up?

3. I did look it up, I'm just having trouble doing it for an undefined bijection

4. Ok, is the relation reflexive? That is, can you find a bijection from a set to itself? Of course you can! The trivial bijection of mapping each element to itself.

Is the relation symmetric? Say you are given a bijection from A1 to A2, is there a bijection from A2 to A1? Yes. A bijection is invertible, and its inverse is also a bijection.

Is the relation transitive? If there is a bijection from A1 to A2, and a bijection from A2 to A3, is there a bijection from A1 to A3? There is indeed, and what function gives this bijection?