Equivalence relation

**Siron** · October 19th 2011, 07:36 AM

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

I don't get started ...

**emakarov** · October 19th 2011, 07:51 AM

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?

**Siron** · October 19th 2011, 09:32 AM

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

**Plato** · October 19th 2011, 10:55 AM

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 $\Delta=\{(x,x):x\in X\}$
Is it true that $\Delta\in(E(X)\cap F(X))~?$ .

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

**Deveno** · October 19th 2011, 02:56 PM

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

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