What does an equivalence class look like?

Question

Let $\mathbb{R}^2 = \{Q = (a,b) | a,b\in \mathbb{R}\}$. Prove that if $q_1 = (a_1,b_1)$ and $q_2=(a_2,b_2)$ are equivalent, meaning $a_1^2+b_1^2 = a_2^2 +b_2^2$, then this gives an equivalence relation on $\mathbb{R}^2$. What is $[(1,0)], [(0,1)],[(2,2)],[(0,0)]?$ What does an equivalence class look like?

I am not sure how to do the second part of the question?

First part proof: Let $q_1$ and $q_2$ be equivalent then:

Reflexive: Let $a\sim b$ then $a \sim a$. So $a_1^2+a_1^2 = a_2^2 +a_2^2$ implies $2a_1^2 = 2a_2^2$ implies $a_1^2 = a_2^2$.

Symmetry: We must show $a\sim b$ and $b\sim a$. Thus let $a\sim b$ then we have $a_1^2+b_1^2 = a_2^2 +b_2^2$ are equivalent thus $b\sim a$ implies $b_1^2+a_1^2 = b_2^2 +a_2^2$ which are equivalent.

Transitive: $a\sim b$ and $b\sim c$ implies $a\sim c$. So $a_1^2+b_1^2 = a_2^2 +b_2^2$ and $b_1^2+c_1^2 = b_2^2 +c_2^2$ thus if we add them we have $a_1^2+2b_1^2+c_1^2 = a_2^2 +2b_2^2+c_2^2$ which implies $a_1^2+c_1^2 = a_2^2 +c_2^2$.

Answer 1

To show that something is an equivalence relation, you need to show that it satisfies:

1. Reflexivity
2. Symmetry
3. Transitivity

I will edit this post as needed. For now, let's focus on:

Reflexive Property

A relation $\sim$ is called reflexive if $a\sim a$ for all $a$ in your set.

In your problem, the set is $\mathbb{R}^2.$ In order for your $\sim$ to be an equivalence relation, no matter which point $a$ I pick from $\mathbb{R}^2$, $a\sim a$ has to be true.

The biggest problem you are having is that you are mixing up the indices and different letters.

Pick a point in $\mathbb{R}^2$. To avoid confusion with indices, let's just call this point $(x,y)$.

Let's verify and see if $(x,y)\sim (x,y)$.

According to the definition you have, $(a_1,b_1)\sim (a_2,b_2)$ means

$$a_1^2 + b_1^2 = a_2^2 + b_2^2$$

See how everything on the left had side of the equality comes from only the first point $(a_1,b_1)$? (And everything on the right hand side of the equality comes from only the second point $(a_2,b_2)$?)

With that in mind, can you rewrite $(x,y)\sim (x,y)$? Think about what should be on the left side of the "=" sign and what should be on the right side of the "=" sign.

Answer 2

A geometric interpretation of each equivalence class is being asked in the second part. What can you say about the sets of points on the plane which are equivalent?

Hint: Look at the distance (or really distance squared) of those points from the origin