What is the Definition of a Relation in Mathematics?

[quantum123]

It is kinda strange. There is no agreement on the definition of a relation.
Some books says it is a set of ordered pairs.
Other books says it is a subset of a cartesian product.
How nice if everything can be agreed down to a few axioms like Euclid's elements.

What is your favourite definition of a relation?

 

[micromass]

The two definitions say the same thing.

 

[quantum123]

I do not agree with that. In the definition using ordered pairs, it is assumed that a set can be built from ordered pairs. But in the cartesian definition, a set is provided already, you just use a part of it via subset.

 

[xxxx0xxxx]

I go with:

$$R \mbox{ is a relation} \Leftrightarrow \forall r(r \in R \Rightarrow \exists x \exists y (r=<x,y>))$$

This definition states that the elements of R are ordered pairs if and only if R is a relation.