What are equivalence classes that would not have (1+1 and 2 ) or (2 x 3 and 6) in same classes? Are there any ordered equivalence classes that can be used to distinguish 1+1 and 2 ,
On one side there is an operation and on the other side just a single number.
This becomes more obvious when dealing with factoring and multiplication , e.g.
17.19 = 323 , LHS contains more information than RHS but how to put an order on them?
 A: An equivalence relation on a set $S$ partitions $S$ into disjoint subsets. What is $S$ in your case?  
If $S$ is a set of standard numbers, like  $\mathbb R$ or $\mathbb N$, then you can't put $1+1$ and $2$ into separate equivalence classes, because they are the same element (of $\mathbb R$ or $\mathbb N$ or whatever).  
But if your set $S$ is the set of strings that represent well-formed arithmetic expressions, then you can define an equivalence relation according to (for example) the number of arithmetic operations in the string. Then "$2$" is equivalent to "$323$", and "$1+1$" is equivalent to "$17 \times 19$".
A: You could look at the set $\mathbb N\times\mathbb N$ of ordered pairs of natural numbers with equality as equivalence relation. The single number $2$ should then be written as $(2,0)$. For this equivalence relation, $(1,1)$ and $(2,0)$ are not equivalent.
On the set $\mathbb N\times\mathbb N$, there is then a weaker equivalence relation defined by: $(a,b)\sim (c,d)$ if and only if $a+b=c+d$.
