Define a relation $ \sim$ on $\mathbb{R}^2 \setminus (0,0)$ by $(a,b)\sim (c,d)$ if there is some real number x with $a=xc$ and $b=xd$.

Question

Define a relation $ \sim$ on $\mathbb{R}^2 \setminus (0,0)$ by $(a,b)\sim (c,d)$ if there is some real number $x$ with $a=xc$ and $b=xd$. I need to prove the relation is an equivalence relation and determine the equivalence classes.

Here's what I have started.

Reflexive: Let $(a,b) \epsilon \sim$, then $a=1\cdot a$ and $b=1\cdot b$. Thus $(a,b)\sim(a,b)$ and $\sim$ is reflexive.

Can I have a nudge to finish the rest?

Even more important though...can I some deeper intuition to their relation? Help with understanding that will help me determine the equivalence classes on my own.

Answer 1

Symmetry comes from the fact that you can divide by $x$ since neither $a$ or $b$ are $0$, and transitivity comes from considering the product of $x_1$ and $x_2$.

For what the equivalence classes are, think of $\mathbb{R}^2$ as the Euclidean plane, and imagine lines through the origin.