Drawing equivalence classes

Question

Define two point $(a, b)$ and $(c, d)$ to be equal if and only if $b^2+a^2=d^2+c^2$. The task at hand was to prove this an equivalence relation, and to describe and sketch its equivalence classes. I am a poor spatial thinker so I'm struggling to understand how to do the latter. Some help would be greatly appreciated!

Answer 1

Hint:

$a^2 + b^2 = c^2 + d^2$ if and only if $\sqrt{a^2 + b^2} = \sqrt{c^2 + d^2}$.

Now, $\sqrt{a^2 + b^2}$ is the distance of the point $(a,b)$ from $(0,0)$, and $\sqrt{c^2 + d^2}$ is the distance of the point $(c,d)$ from $(0,0)$.

So $(a,b)$ is equivalent to $(c,d)$ means what?

Answer 2

I sympathize with you, I hate spatial reasoning and avoid it like the plague so here is my approch. As the former is fairly trivial and I assume by your post you have already done it I will go on the latter entirely. Let us write $\sim$ for the relation and we have that $(a,b)\sim(c,d)$ if and only if $a^2+b^2=c^2+d^2$. Notice that the collection of equivalence classes is written as $\mathbb{R}^2/\sim$

Proposition: There exists a bijection $\varphi:\mathbb{R}^2/\sim\to\mathbb{R}_{0\leq}$.

We define $\varphi:\mathbb{R}^2/\sim\to\mathbb{R}_{0\leq}$ by $\varphi((a,b))=a^2+b^2$. We need to check that this is well-defined for completeness, let $(a,b)\sim(c,d)$ and then we have $$\varphi((a,b))=a^2+b^2=c^2+d^2=\varphi((c,d))$$ We will check if it is surjective injective. For surjectivity, let $(a,b)$ be given and $\varphi((a,b))=a^2+b^2=\epsilon$. Then we pick $|a|\leq\sqrt{\epsilon}$ and set $b=\pm\sqrt{\epsilon-a^2}$ and it is satisfied. Injectivity is fairly obvious by the definitions so it is clearly a bijection.

That means we have an inverse and we can write it as $\sigma(r)=\{(\pm a,\pm\sqrt{r-a^2})\in\mathbb{R}^2:a\leq\sqrt{r}\}$ This will give us all equivalence classes, of course any equivalent writing would be just as much correct.

Personally I prefer just prefer, as we are not talking about any specific structure, that the equivalence classes are equivalent to the non-negative real numbers.