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Thread: Equivalence relation

  1. #1
    MHF Contributor Siron's Avatar
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    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 $\displaystyle E(X)\cap F(X)$.

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  2. #2
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    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?
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  3. #3
    MHF Contributor Siron's Avatar
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    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.
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    Re: Equivalence relation

    Quote Originally Posted by Siron View Post
    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 $\displaystyle g\in(E(X)\cap F(X))~?$
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  5. #5
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    Re: Equivalence relation

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