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Math Help - Equivalence relation and Functions

  1. #1
    Junior Member mremwo's Avatar
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    Equivalence relation and Functions

    The problem is:
    "Suppose that A is a nonempty set and R is an equivalence relation on A. Show that there is a function f with A as its domain such that (x,y) are elements of R if and only if f(x)=f(y)"

    I don't understand how to connect a relation with a function

    thank you
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by mremwo View Post
    The problem is:
    "Suppose that A is a nonempty set and R is an equivalence relation on A. Show that there is a function f with A as its domain such that (x,y) are elements of R if and only if f(x)=f(y)"

    I don't understand how to connect a relation with a function

    thank you
    A function is actually a relation, but that's just not how many people think about it. Consider that a function is really just it's graph \Gamma_f=\left\{(x,f(x)):x\in X\right\}. This \Gamma_f is the relation.
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  3. #3
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    Try finding a function f:A\rightarrow \mathcal{P}(A) (the power set of A) such that two elements are in the same equivalence class under R if and only if they have the same image under f. Can you think of a natural choice for the image under f of an element from A?
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by topspin1617 View Post
    Try finding a function f:A\rightarrow \mathcal{P}(A) (the power set of A) such that two elements are in the same equivalence class under R if and only if they have the same image under f. Can you think of a natural choice for the image under f of an element from A?
    What does this have to do with anything? Unless you really meant 2^{A\times A}.
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  5. #5
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    No. The function f:A\rightarrow \mathcal{P}(A) given by x \mapsto [x] does exactly what the question asks.
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by topspin1617 View Post
    No. The function f:A\rightarrow \mathcal{P}(A) given by x \mapsto [x] does exactly what the question asks.
    Touche sir, touche. I misread the question.
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