Equivalence Classes for a Cartesian Plane Relation

**sritter27** · March 5th 2009, 09:39 PM

Greetings,

If a relation is defined on the Cartesian plane $\mathbb{R} \times \mathbb{R}$ by $(x_1,y_1) \; \backsim \; (x_2,y_2) \; \Longleftrightarrow \; (x_2,y_2) \; = \; (rx_1,ry_1)$ for some $r \; > \; 0$ , what would the equivalence classes be? This seems like it should be easy, but I'm drawing a blank on it for whatever reason so for any help I would be grateful.

**Jhevon** · March 5th 2009, 10:00 PM

Originally Posted by sritter27

Greetings,

If a relation is defined on the Cartesian plane $\mathbb{R} \times \mathbb{R}$ by $(x_1,y_1) \; \backsim \; (x_2,y_2) \; \Longleftrightarrow \; (x_2,y_2) \; = \; (rx_1,ry_1)$ for some $r \; > \; 0$ , what would the equivalence classes be? This seems like it should be easy, but I'm drawing a blank on it for whatever reason so for any help I would be grateful.

the equivalence class of the point $(x_1,y_1)$ is the set of all points on the line with slopes of $\frac {y_1}{x_1}$ passing through the origin

can you show this?

**sritter27** · March 5th 2009, 10:25 PM

Originally Posted by Jhevon

the equivalence class of the point $(x_1,y_1)$ is the set of all points on the line with slopes of $\frac {y_1}{x_1}$ passing through the origin

can you show this?

I think I can see why that would be an equivalence class, but is there anything stopping $\frac {x_1}{y_1}$ from being one as well? Perhaps I'm over-thinking it or don't fully understand equivalence classes as well as I should.

**Jhevon** · March 5th 2009, 10:31 PM

Originally Posted by sritter27

I think I can see why that would be an equivalence class, but is there anything stopping $\frac {x_1}{y_1}$ from being one as well? Perhaps I'm over-thinking it or don't fully understand equivalence classes as well as I should.

slope is given by $\frac {y_2 - y_1}{x_2 - x_1}$

the equivalence class of some element on a set, is the set of all other elements in the set that relate to the said element under an equivalence relation.

so given any random point, it will relate to all points on the line i mentioned. that is, they will be equivalent under the given relation.

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