Can someone explain how to find the number of equivalence classes and elements? I am struggling so much with this topic. Trying to do some practice questions but I don't seem to get it. What I'm working on is

Let $A = \{ 1, 2,  3, \dots,  2014 \} = \{ x \mid 1  \le x  \le 2014 \}$. 
   Let $P$ be the set of all non-empty
  subsets of $A$. Define the relation R on P by: 
  for any $X, Y$ element of $P$, $XRY$ if and only if the largest element of $X$ equals the largest element of $Y$.

(b) List all the elements of the equivalence class $[\{3\}]$ (the equivalence class of $\{3\}$).
(c) How many equivalence classes does $R$ have? Explain.
(d) How many elements does the equivalence class $[\{271\}]$ (the equivalence class of $\{271\}$)
have? Explain.
 A: For (b), you must find all non-empty subsets of $A$ whose largest element is $3$
For (c), consider the function $\max : P \mapsto A$, which takes a subset to its maximum. By definition $X R Y$ iff $\max(X) = \max(Y)$. So every equivalence class corresponds to an element of $A$.
For (d), how may subsets are there that have $271$ as their maximum? If $A$ is such a set, then $A$ is the disjoint union of $\{ 271 \}$ and $B$, where $B$ is an arbitrary subset of $\{1, \dots, 270 \}$.
A: for $[\{3 \}]$ You are looking for all the sets that are subsets of $A$ that have $3$ as their largest element and so 
$$[\{3\}] = \{\{1,2,3\},\{1,3\},\{2,3\},\{3\}\}$$
Notice that we don't include the empty set
How many equivalence classes does $R$ have ?
$R$ has $2014$ different equivalence classes, because we have $2014$ different numbers . And actually all these equivalence classes are distinct and they all partition the set $P$
I will let you think about (c) and ask me if you couldn't get it
Just look at the equivalence class of ${3}$ and try to make connections 
To make it easier for you consider $$[\{4\}] = \{\{1,2,3,4\},\{2,3,4\},\{1,2,4\},\{1,3,4\},\{2,4\},\{3,4\},\{1,4\},\{4\}\}$$
Now count the elements in the equivalence class of {3} and {4} and try to drive a formula for 271
