Relation theory

Question

Let S be a set and R a relation on that set. A subset T of S is said to be a right R-set if it is of the form {x|sRx} for some constant s in S. The collection of all right R-sets is a subset of P(S), the powerset of S. My question is, for any nonempty set S, and for any nonempty subset X of P(S), such that the cardinality of X is not greater than the cardinality of S, is there a relation R on S such that X is the collection of right sets of R?

Answer 1

So we have $S$ and we have $\mathcal{X} \subseteq \mathbb{P}S$, where $|\mathcal{X}|\leq|S|$, and we want an $R$ with $$\mathcal{X} = \{ \{x : s \ R \ x \} : s \in S\}$$

Since we have access to $\mathcal{X}$, our relation $R$ will most likely depend on it.

Since $|\mathcal{X}|\leq|S|$, we can find a surjection $f : S \to \mathcal{X}$.

Now, the relation $$s \ R \ x \ :\equiv\ x \in f(s)$$ ought to do the trick :)