I'm having hard time understanding the difference between (binary) relation and correspondence.
The definition of binary relation that I know is:
A (binary) relation R between sets X and Y is a subset of the Cartesian product X x Y.
Example with the relation "=": $$X=\{0,1,2,3\}, Y=\{0,1,2,3\}$$ $$R = \{(0,0), (1,1), (2,2), (3,3)\}.$$
The definition of correspondence that I know is:
A correspondence f between X and Y is a triple (X,Y,R) where R is a subset of the Cartesian product X x Y.
Example for correspondence: $$X=\{0,1,2,3\}, Y=\{0,1,2,3\}$$ $$f = (\{0,1,2,3\}, \{0,1,2,3\}, \{(0,0), (1,1), (2,2), (3,3)\})$$
Here is suggested that correspondence is the old term for relation, but I'm still not very convinced, because of the examples that I gave above. Any thoughts?
There is little difference. Correspondances package the definition into a triplet while relations rely upon a verbal description which can have the disavantage of loosing sight of X and Y.