Proving $|S/R^2|=\aleph$ , $(x_1,x_2)S(y_1,y_2)\iff x_1^2-x_2=y^2_1-y_2$

Question

Let $S$ be an equivalence relation over $\mathbb R^2$ such that: $(x_1,x_2)S(y_1,y_2)\iff x_1^2-x_2=y^2_1-y_2$

Prove that $|S/R^2|=\aleph$

One side is pretty simple: $|S/R^2|\le |\mathbb R^2|=\aleph$.

We're left with showing $|S/R^2|\ge \aleph$, I thought of defining an injective function from $S$ to $\mathbb R$ like so:

$f((x_1,x_2),(y_1,y_2))=2^{x_1}3^{x_2}5^{y_1}7^{y_2}$

Is this the right way?

How does one of the equivalence classes look like here?

Answer 1

First let me point out two notational issues:

1. It is important to be consistent. Don't write $R^2$ when you mean $\Bbb R^2$, or vice versa, and have used the same notation before in the same sentence.

2. The usual notation is $A/E$ for the quotient set, not $E/A$. So you would in fact want $\Bbb R^2/S$ rather than $S/R^2$.

While the trick of taking powers of prime numbers is a very good trick for showing that $\Bbb N^4$ and $\Bbb N$ have the same cardinality, it doesn't work for $\Bbb R$ since the function is continuous and there are no continuous injections from $\Bbb R^4$ into $\Bbb R$.

Additionally this is the wrong approach, since it establishes that $S$ and $\Bbb R$ have the same cardinality, not that $\Bbb R^2/S$ and $\Bbb R$ have the same cardinality.

What you should do, is understand what the equivalence relation mean, and find a way to pick a canonical representative from each equivalence class. For example, what happens if $x_2=0$? How many values could $x_1$ assume within the same equivalence class?

Now use that in order to define an injection from a subset of $\Bbb R$ into $\Bbb R^2/S$, namely find some $A\subseteq\Bbb R$ such that $|A|=|\Bbb R|$, and the map $a\mapsto a/S$ is injective (so no two elements of $A$ have the same equivalence class).