.In my topology course I have encountered with Schroeder-Bernstein theorem and before it I have dealt with Numerical equivalence.
It says "Numerical equivalence has all the properties of an equivalence relation, and if we consider it as an equivalence relation in the class of all non-empty subsets of some universal set $U$ , it groups together into equivalence sets all those subsets of $U$ which have the same number of elements.
Could any one explain the above assertion it's really confusing me. Also explain how this result relates to Schroeder-Bernstein theorem and explain the theorem in simple terminology.
Here are the axioms of an equivalence relation, and the axioms of a partial order and preorder are here for use shortly. (While written for sets, it is still possible to discuss the axioms given the context of your universal set.)
Equipotence is an equivalence relation
Equipotence is trivially an equivalence relation on the class of sets. It satisfies all the axioms. Check them.
Every equivalence relation partitions its underlying set into nonoverlapping pieces, within which everything is mutually related. It helps to think of these as "big points" made up of collections of points.
Schroder-Bernstein
The connection with the Schroeder-Bernstein theorem is the preordering of the class of sets this way: define $A\leq B$ if there is an injective function $A\to B$. It is obvious that this is a preorder: reflexivity and transitivity are easily verified.
Now, there is no way to prove that if $A\leq B$ and $B\leq A$, then $A=B$ as sets, because there exist disjoint, equipotent sets. Thus, this preorder is not a partial order on the class of sets. However, after deciding to identify each set with all sets equipotent to it, you do have a partial order (on the equivalence classes of mutually equipotent sets.)
If we write $|A|$ to indicate the equipotence class of the set $A$, we can migrate our preorder to the classes by saying $|A|\leq |B|$ if there exists an injective set function from $A\to B$. It's still obviously a pre-order, but then the Schroeder-Bernstein theorem is literally the proof that this $\leq$ is antisymmetric, and hence a partial order on the equipotence classes.