Let A be a finite set of size k and R a relation on the...

Step 1
There are three criteria that must be followed in order to prove R is an equivalence relation,
1. Reflexive
2. Symmetric
3. Transitive
Step 2
To get you started, with generic sets $B,C,D\in \mathcal{P}(A)$, see that we always have $|B|=|B|$
so R is reflexive. This would mean $(B,B)\in R$ for some $B\subseteq A$
To show it's symmetric, it requires you to show that if $|B|=|C|$, then $|C|=|B|$.
To show it's transitive, it requires you to show if $|B|=|C|$, and $|C|=|D|$, then $|B|=|D|$

Step 1
Informally, under this equivalence relation two subsets are equivalent when they have the same size.
Thus, the equivalence class of {a} consists of all subsets of A with cardinality/size equal to one.
Thus the size of this equivalence class is $k=|A|$.
Step 2
The equivalence class of {a,b} consists of all two element subsets of A. Thus the size of this equivalence class is $(\genfrac{}{}{0ex}{}{k}{2})=\frac{k(k-1)}{2}$