Help Finding the Size of a Quotient Set

Question

I am having difficulties solving the following question.

Given a finite set $X$ and $S \subseteq X$. $R$ is the equivalence relation over $P(X)$ defined as: $$(A, B)\in R \iff A \cup S = B \cup S$$ Find the size of $P(X)/R$.

I have found that $[A]_R = \{B \in P(X) \lvert S' \in P(S), B = A \cup S'\}$ and I know that $P(X)/R = \{[A]_R \lvert A \in P(X)\}$. I am struggling to find the size of $[A]_R$ and the size of the quotient set.

Answer 1

$(A,B)\in R\iff A\setminus S=B\setminus S$. This means that $(A,B)\in R$ if and only if $A$ and $B$ have the same elements oustside of $S$ (which is what you already found, except that it should be $[A]_R=\{ B\in\mathcal{P}(X),\exists S'\subseteq S,B=A\cup S' \text{ or } A=B\cup S' \}$). Now let $\varphi:\mathcal{P}(X\setminus S)\rightarrow\mathcal{P}(X)/R$ defined by $\varphi(A)=[A]_R$. $\varphi$ is surjective because for all $[A]_R\in\mathcal{P}(X)/R,[A]_R=\varphi(A\setminus S)$ with $A\setminus S\in\mathcal{P}(X\setminus S)$. $\varphi$ is injective because if $A,B\in\mathcal{P}(X\setminus S)$ are such that $[A]_R=[B]_R$, then because of what said above $A\setminus S=B\setminus S$ i.e. $A=B$. Therefore $\varphi$ is a bijection and $\mathcal{P}(X)/R\simeq\mathcal{P}(X\setminus S)$.