I have very well understood the difference between Comparable and Comparator interfaces, and
as such the ordering imposed by them.

Further, i am clear on why the compareTo must be consistent with equals method.
From [Oracle docs](http://docs.oracle.com/javase/7/docs/api/java/lang/Comparable.html), 

> 
> It is strongly recommended (though not required) that natural orderings be consistent
>  with equals. This is so because sorted sets (and sorted maps) without explicit 
>  comparators behave "strangely" when they are used with elements (or keys) 
>  whose natural ordering is inconsistent with equals. In particular, such a 
>  sorted set (or sorted map) violates the general contract for set (or map), 
>  which is defined in terms of the equals method.
> 
> 
> For example, if one adds two keys `a` and `b` such that 
>  `(!a.equals(b) && a.compareTo(b) == 0)` to a sorted set that does not use an 
>  explicit comparator, the second add operation returns false (and the size of 
>  the sorted set does not increase) because a and b are equivalent from the 
>  sorted set's perspective.
> 
> 
> 

However, I am unable to clear myself on the paragraph(below),

> 
> For the mathematically inclined, the *relation* that defines the natural ordering on 
>  a given class C is:
> 
> 
> 
> ```
>     {(x, y) such that x.compareTo(y) <= 0}.
> 
> ```
> 
> The *quotient* for this total order is:
> 
> 
> 
> ```
>     {(x, y) such that x.compareTo(y) == 0}.
> 
> ```
> 
> It follows immediately from the contract for `compareTo` that the quotient is an
>  *equivalence relation* on C, and that the natural ordering is a total order on C. 
>  When we say that a class's natural ordering is *consistent with equals*, we mean 
>  that the quotient for the natural ordering is the equivalence relation defined 
>  by the class's `equals(Object)` method:
> 
> 
> 
> ```
>     {(x, y) such that x.equals(y)}.  
> 
> ```
> 
> 

What is the author trying to suggest here?. Can someone explain with a simple example.