What does "the partition of $\mathcal{P}(A)$ induced by $R$" mean?

Question

I am working on the following question:

Let $A$ be a non-empty set and fix a subset $B$ of $A$. Define a relation $R$ on the set $\mathcal{P}(A)$ of subsets of $A$ as follows: $$R = \{(X, Y) | X \cap B = Y \cap B \},$$ where $X$ and $Y$ are subsets of $A$.

a) Show that $R$ is an equivalence relation.

b) If $A = \{ 1, 2, 3 \}$ and $B = \{ 1, 2 \}$ find the partition of $\mathcal{P}(A)$ induced by $R$.

c) If $A = \{ 1, 2, 3, 4, 5 \}$ and $B = \{ 1, 2, 3 \}$ find the equivalence class $[X]$ if $X = \{ 1, 3, 5 \}$.

I couldn't understand the meaning of the part (b). What does it mean?

Answer 1

If $A = \{1,2,3\}$, then $\mathcal P(A) = \{\varnothing,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},A\}$. Now, for every $X \in \mathcal P(A)$ (the eight possible choices) write explicitly the elements of the set $$[X]_R := \{Y \in \mathcal P(A) : (X,Y) \in R\} = \{Y \in \mathcal P(A) : X \cap \{1,2\} = Y \cap \{1,2\}\}.$$ Now, the partition of $\mathcal P(A)$ induced by $R$ is the set given by $\{[X]_R : X \in R\}$.