Let $A$ be a nonempty set and let $B$ be a fixed subset of $A$. A relations $R$ is defined on the power set $\mathcal{P}(A)$ by $X\mathrel{R}Y$ if $X \cap B = Y \cap B$.
Let $A=\{1,2,3,4,5\}$ and $B=\{1,3\}$. For a subset $X=\{2,3,4\}$, Determine the equivalence class $[X]$?
I have never heard of something like this. Any ideas to get started?
Note $[X]=\{Y \subset A \mid X\mathrel{R}Y \}$. You have that $X \cap B=\{3\}$, thus \begin{equation*} [X]=\{Y \subset A \mid Y \cap \{1,3\} = \{3\}\} \end{equation*} What other subsets of $A$ can you construct that intersect with $B$ to get $\{3\}$? Some examples include $\{3,4\}$ and $\{2,3,5\}$.