We can define a binary relation as a set of ordered pairs. Alternatively, we can call the set of ordered pairs the "graph" of the relation, and define the relation itself as a triple $(X,Y,f)$, where $f$ is the graph, $X$ is its domain and $Y$ is its codomain. The same goes for functions; they can be defined as sets of ordered pairs, or as triples. So my question is, given that its more lightweight to define a binary relation simply as its graph, why is the ordered-triple approach much more common?
Here's a list of reasons and possible objections that I've come up with.
Reason 0: The ordered-triple approach allows us to distinguish between surjections and non-surjections. Possible Objection: This could be read another way, as suggesting that surjectivity is a contrived concept. Maybe we should only every say "$f \in \mathrm{Surj}(X,Y)$" but never simply say "$f$ is a surjection."
Reason 1: Category theory works that way. Possible Objection: Perhaps this is a "hint" that maybe there's room for improvement in the basic definitions of category theory.
Reason 2: The ordered triple approach allows us to define the complement of a relation by $f^c = X \times Y \setminus f$. Thus, the set of all relations with domain $X$ and codomain $Y$ form a Boolean algebra. Possible Objection: This is a pretty minor advantage, given that we can just write "Defining $A^c = X\times Y \setminus A$, it follows that..." whenever we need to.
So my question is, what's the payoff of the ordered triple approach?