# Proof regarding Equivalence Relations

• Dec 9th 2012, 08:28 AM
MachinePL1993
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?
• Dec 9th 2012, 08:48 AM
Plato
Re: Proof regarding Equivalence Relations
Quote:

Originally Posted by MachinePL1993
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?
• Dec 9th 2012, 08:58 AM
MachinePL1993
Re: Proof regarding Equivalence Relations
Yes, $R,S$ are in the same domain. $RS$ is a composition of two relations.
• Dec 9th 2012, 09:38 AM
Plato
Re: Proof regarding Equivalence Relations
Quote:

Originally Posted by MachinePL1993
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
$\Delta_X\subseteq T,~T=T^{-1},~\&~T\circ T\subseteq T$.