Exercise 3, § 6

Let E be a set, A a subset of E, and R the equivalence relation associated with the mapping $\mathrm{X} \rightarrow \mathrm{X} \cap \mathrm{A}$ of $\mathfrak{P}(\mathrm{E})$ into $\mathfrak{P}(\mathrm{E})$. Show that there exists a bijection of $\mathfrak{P}(\mathrm{A})$ onto the quotient set $\mathfrak{P}(\mathrm{E})/\mathrm{R}$.