1. ## Equivalence relation

Let $\displaystyle X$ be a set, $\displaystyle E(X)$ the set of all the equivalence relations on $\displaystyle X$ and $\displaystyle F(X)$ the set of all the functions on $\displaystyle X$. Find $\displaystyle E(X)\cap F(X)$.

I don't get started ...

2. ## Re: Equivalence relation

Suppose R ∈ E(X) ∩ F(X). Then R is an equivalence relation and a function. Suppose (x, y) ∈ R, i.e., R(x) = y. What can you say about y?

3. ## Re: Equivalence relation

If R has to be an equivalence relation and a function at the same time I think y has to be unique.

4. ## Re: Equivalence relation

Originally Posted by Siron
If R has to be an equivalence relation and a function at the same time I think y has to be unique.
Suppose that $\displaystyle \Delta=\{(x,x):x\in X\}$
Is it true that $\displaystyle \Delta\in(E(X)\cap F(X))~?$.

Suppose that $\displaystyle g\in F(X)$ and $\displaystyle \left( {\exists a \in X} \right)\left[ {g(a) \ne a} \right]$ is it possible that $\displaystyle g\in(E(X)\cap F(X))~?$

5. ## Re: Equivalence relation

consider a slightly easier question: suppose R is merely reflexive, and also a function. how many reflexive functions are there?