$A$ is a non-empty set and $B$ is a fixed subset of A. The relation R on the set of subsets of A as follows. Relation is defined by: $R = \{(X,Y) | X \cap B = Y \cap B\}$ where $X$ and $Y$ are subsets of $A$.
I can show that this relation is an equivalence relation. However, the question also asks me the following: If $A = \{1,2,3\}$ and $B = \{1,2\}$ find the partition of the set of subsets of $A$ induced by $R$.
I could not really understand the expression. Textbook says that equivalence classes of $R$ form a partition of $S$. Should I check every subset of $A$ and find their equivalence classes?
Yes your idea is the good one. Take any subset of $ X \subseteq A$ (there is only 8 of those) and compute $X \cap B$ (easy!). Then look at the values of $X \cap B$ to find the equivalence classes.
You’ll find
$\overline{\{1\}}=\{\{1\}, \{1,3\}\}$, $\overline{\{2\}}=\{\{2\}, \{2,3\}\}$, $\overline{\emptyset}=\{\emptyset, \{3\}\}$, $\overline{\{1,2\}}=\{\{1,2\}, \{1,2,3\}\}$