Set of Equivalence relations over $\mathbb{R}$

Question

I was taught that the cardinality of equivalence relations in $\mathbb{R} \to \mathbb{R}$ is $2^\mathfrak{c}$.

I was wondering if it could be proven using CSB.

Any help would be appreciated.

Answer 1

Given a subset $S$ of $\mathbb R\setminus\{0\}$, you can define an equivalence relation $\sim$ on $\mathbb R$ by $$a\sim b\iff(a\in S\land b\in S)\lor(a\notin S\land b\notin S)$$ for any $a,b\in\mathbb R$. Note that exclusion of $0$ ensures that the maping is injective (otherwise a set and its complement would lead to the same equivalence relation), but doesn't affect the cardinality. Therefore there exist at least as many equivalence relations as there exist subsets of $\mathbb R$.

On the other hand, an equivalence relation on $\mathbb R$ is a relation on $\mathbb R$, and thus a subset of $\mathbb R\times\mathbb R$. And $|\mathbb R\times \mathbb R|=|\mathbb R|$. Thus there exist at most as many equivalence relations as there exist subsets of $\mathbb R$.

Answer 2

I'm assuming CSB is Cantor-Schröder-Bernstein.

One family of $2^c$ equivalence relations in $\mathbb R$ is this. For each subset $S$ of $[0,1]$, take $x \sim x+1$ for $x \in S$.

On the other hand, any relation in $\mathbb R$ can be considered as a subset of $\mathbb R \times \mathbb R$.