How do you define a relation on R x R = R^2 by (x,y) ≡ (a,b) iff x+y = a+b?

Question

Given questions:

a) Prove that ≡ is an equivalence relation

b) Describe the equivalence classes geometrically. (Remember the equivalence classes in this case will be a partition of the plane)

I appreciate your feedback on helping me in part (b) and feel free to comment.

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I already did the equivalence relation and checked the three properties: symmetry, reflexivity and transitivity. I'm stuck at part b) and you can see in the image that I got some work done in part a). I don't know how to describe it geometrically. I've used a bunch of number examples to see the behavior of some (x,y) ≡ (a,b)

Answer 1

Let's try to imagine a single equivalence class. Let's say $(x_0, y_0)$ is a member of this class, $[(x_0,y_0)]$. What other points does the class conclude?

Well, by definition, the equivalence class is the set of all points that are in relation to $(x_0, y_0)$, so

$$[(x_0, y_0)] = \{(x,y)| (x,y)\equiv (x_0, y_0)\}$$

But we can write $\equiv$ by using its definition, so

$$[(x_0, y_0)] = \{(x,y)| x+y = x_0 + y_0\}$$

A simple reading now tells you that the equivalence class of $(x_0, y_0)$ is simply the line $y = x + (x_0 + y_0)$.

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So, the relation splits the space $\mathbb R^2$ into a set of paralel lines, each line going at an angle of $45$ degrees.