A question about definition of "Binary Relation"

Hello everybody.

Is the following definition true?

A binary relation from a set A to a set B is a subset of Cartesian product of A and B, such that in all ordered pairs *there is or there is not* a certain connection between the first elements (that come from A)and the second elements(that come from B). And the names of the Relations (for example the relation "greater than") describes the connection between first and second elements of ordered pairs.

If it's a true definition, then introduce a source that contains it. And if is not true, describe why.

Thanks.