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Math Help - Proof regarding Equivalence Relations

  1. #1
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    Proof regarding Equivalence Relations

    Hello,
    I have to proof that when R and S are equivalence relations then  R \cup S is an equivalence relation if and only if  R \cup S=RS.

    Can anyone give me a hint?
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  2. #2
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    Re: Proof regarding Equivalence Relations

    Quote Originally Posted by MachinePL1993 View Post
    Hello,
    I have to proof that when R and S are equivalence relations then  R \cup S is an equivalence relation if and only if  R \cup S=RS.
    Can anyone give me a hint?

    You need to expand a bit on notation.
    I assume that R~\&~S have the same domain.

    But the notation RS is not standard. How is it defined?
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  3. #3
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    Re: Proof regarding Equivalence Relations

    Yes,  R,S are in the same domain. RS is a composition of two relations.
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  4. #4
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    Re: Proof regarding Equivalence Relations

    Quote Originally Posted by MachinePL1993 View Post
    Yes,  R,S are in the same domain. RS is a composition of two relations.

    [tex]R\circ S [/tex] gives  R\circ S
    That is the standard notation for composition.

    HINT: if T is an equivalence relation then
    \Delta_X\subseteq T,~T=T^{-1},~\&~T\circ T\subseteq T.
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