# Thread: Equivalence Relations - HELP!!!

1. ## Equivalence Relations - HELP!!!

Ok so i'm a little stuck on how to prove this..

Consider the set of points (x,y) in the plane (real numbers)^2. Define a relation R(subset 2) on the set as follows:

(x1,y1)R(subset 2)(x2,y2) whenever x1y1=x2y2

Prove that R(subset 2) is an equivalence relation.

Any help would be greatly appreciated. I know i need to prove that it is reflexive, symmetric and transitive but i'm not sure on how to go about it.

2. $\begin{gathered}
\left( {x,y} \right)R\left( {x,y} \right) \Leftrightarrow xy = xy\,\,\text{reflexive} \hfill \\
\left( {x,y} \right)R\left( {y,x} \right) \Leftrightarrow xy = yx\,\,\text{symetric}\hfill \\ \end{gathered}$

$
\begin{gathered}
\left( {x,y} \right)R\left( {u,v} \right)\,\& \,\left( {u,v} \right)R\left( {w,z} \right) \hfill \\
xy = uv\,\& \,uv = wz \hfill \\
xy = wz \hfill \\
\left( {x,y} \right)R\left( {w,z} \right) \hfill \\
\end{gathered}

$