# Thread: Proof regarding Equivalence Relations

1. ## Proof regarding Equivalence Relations

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

Can anyone give me a hint?

2. ## Re: Proof regarding Equivalence Relations

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

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

But the notation $\displaystyle RS$ is not standard. How is it defined?

3. ## Re: Proof regarding Equivalence Relations

Yes, $\displaystyle R,S$ are in the same domain. $\displaystyle RS$ is a composition of two relations.

4. ## Re: Proof regarding Equivalence Relations

Originally Posted by MachinePL1993
Yes, $\displaystyle R,S$ are in the same domain. $\displaystyle RS$ is a composition of two relations.

$$R\circ S$$ gives $\displaystyle R\circ S$
That is the standard notation for composition.

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