Need help to understand equivalence class This is in my note
Let S={1,2,3,4} Let R be the relation on P(s) defined by xRy <=>|x|=|y|
how many equivalence classes are there ? 
5
[∅]={∅}
[{2}]={{1},{2},{3},{4}}
[{2,3}]={{1,2},.......... 
[a]=the set of all element of A that are related to A
[{!}]=the set of all elements of P(s) that are related to {1}.
how do you find the number on the left [{2,3}],[∅]???
If you can explain equivalence class to me in other way. ty!
 A: I think, the particular classes written $[\{2\}]$ and $[\{2,3\}]$ are only examples. The main point is, I guess, clear, an $n$ element subset of $S$ is related exactly to the $n$ element subsets by $R$.
In general, if $M$ is a set and $R$ is an equivalence relation on $M$, then the quotient set $M/R$ which formally consists of the equivalence classes, can also be viewed as the realization of having all the elements of $M$ with the original equality replaced by the relation $R$. So that, each element $m\in M$ is (determines) an element in $M/R$, and $m=m'$ holds in $M/R$ iff $\ m\,R\,m'$ in $M$. Formally, to distinguish, we should rather write it using brackets, like $[m]=[m'] \iff m\,R\,m'$.
Another important perspective is equivalence relations via surjections. Every equivalence relation on $M$ can be defined by a (surjective) function $f:M\to K$ onto some set $K$: 

Let $x\,E_f\,y$ iff $\ f(x)=f(y)$.

Then, this same $f$ determines a bijection between $M/E_f$ and $K$, in other words, in this case the quotient set can be identified with the range of $f$ ($K$), via the mapping $f$, so that each value of $K$ will straightly correspond to one equivalence class.
In this particular example, for $A\in P(S)$, we can set $f(A):=|A|$, then $R=E_f$, and now it takes values from the range $\{0,1,2,3,4\}$.
A: I think you might fare better if you step away from the problem at hand and try to understand equivalence classes. To do this, you must understand that equivalence classes come from equivalence relations. So you should probably start there. Really, you have to take the time to digest the formal definition, no one can do it for you.
A key fact is that an equivalence relation partitions a set. A partition of a set $X$ is a decomposition of $X$ into nonempty sets with the property that these sets do not overlap and that the union of the sets is $X$.
When we recognize that equivalence relation partitions a set, we can identify an equivalence class by the sets that make up the partition. This provides an intuitive idea as to what an equivalence class is. Take the positive integers $\mathbb{N}$ for example. We can partition by divisibility by 2: $x$ is related to $y$ if $x-y$ is divisible by 2. In this case, the set of even numbers is an equivalence class and the set of odd numbers is an equivalence class. 
Back to your example, what is $P(S)?$ For you equivalence relation, what does it mean if $x,y\in P(S)$ and $|x|=|y|$? These are the easy questions.
It seems to me that you are trying to understand the problem before you understand equivalence classes. 
