# 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 \$\displaystyle R \cup S\$ is an equivalence relation if and only if \$\displaystyle 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 \$\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?
• Dec 9th 2012, 08:58 AM
MachinePL1993
Re: Proof regarding Equivalence Relations
Yes, \$\displaystyle R,S\$ are in the same domain. \$\displaystyle 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, \$\displaystyle R,S\$ are in the same domain. \$\displaystyle RS\$ is a composition of two relations.

[tex]R\circ S [/tex] 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\$.