How Do Sets A and B Relate Through Equivalence Properties?

[pairofstrings]

How to define relationship between two sets (mathematics)?
Please, let me make an assumption here.
Assumption : We have a set 'A' and it has a equivalent relation with another set 'B'.

Now, remember that for a set to have an equivalent relation with itself or another set it should satisfy three properties of a set i.e., 1) Reflexive property 2) Symmetric property and 3) Transitive property.
But first in order for these properties to hold true we must first validate that f c= AxB.
Now, let's keep this properties aside.

My question to you is :

1. What is it that is happening between two sets that set 'A' has to map to a particular element is set 'B'.
2. Is the set 'A' related to set 'B' by some function which has a dependent and independent variable? If 'yes' how to determine that function?
OR
3. Only properties (described above) of both the sets will determine whose elements will map to whom. If 'yes' then how can only properties decide the mapping between two sets? Explain.

Please help.
Thank you.

 

[Stephen Tashi]

Is isn't clear what you are asking. There is more than one way to define an equivalence relation on sets. You might be asking about the equivalence relation defined by "A has the same cardinality as B". It is correct that this equivalence is established by defining a 1 to 1 mapping between the elements of A and the element of B. This mapping does not have to be done in any particular way and there may be more than one way to do it. There merely has to be some way to do it.