The question is a follow up of this one.
Denote by $\mathcal P(\mathbb R)$ the power set of the reals and $A$ the subset of $\mathcal P(\mathbb R)$ consisting of the real subsets $X$ such that $x,y \in X$, $x \neq y$ implies $x-y \in \mathbb R \setminus \mathbb Q$.
Is there an uncountable element of $A$? Can such an element be described explicitly (I know that this question lacks precision)? What is the cardinality of $A$?
Define a relation $\sim$ on $\Bbb R$ by $x\sim y$ iff $x-y\in\Bbb Q$. It’s easy to verify that $\sim$ is an equivalence relation. For $x\in\Bbb R$ let $[x]$ be the $\sim$-equivalence class of $x$. Then $[x]=x+\Bbb Q=\{x+q:q\in\Bbb Q\}$, so $[x]$ is countable. Let $\mathscr{C}=\{[x]:x\in\Bbb R\}$; $\Bbb R$ is uncountable, so $\mathscr{C}$ must also be uncountable. In fact
$$|\Bbb R|=|\mathscr{C}|\cdot|\Bbb Q|=\max\{|\mathscr{C}|,|\Bbb Q|\}=|\mathscr{C}|\,,$$
so we can go further and say that $|\mathscr{C}|=|\Bbb R|$. Now let $C$ contain exactly one member of each $\sim$-equivalence class; then $|C|=|\mathscr{C}|=|\Bbb R|$, and $C\in A$. Every subset of $C$ is also in $A$, so $A$ contains subsets of $\Bbb R$ of every possible size, i.e., every cardinality not exceeding $|\Bbb R|$.