[Math] Can only one ordered pair be a relation


I'm sorry, but I really can't find an answer to this no matter how deep I dig.

A relation is defined as any set of ordered pairs.

But what about a set of only one ordered pair? Is it still a relation? Is it a special kind of relation? I get Binary Relation in some pages but is a Single,Singular,Just-one-ordered-pair relation an actual thing?

I know it's quite elementary but thanks!

Best Answer

A binary relation (or relation, means the same) from a set $A$ to a set $B$ is any subset $R\subseteq A\times B$. We take any here seriously so in particular, if $A$ contains some element $a$, and $B$ contains some element $b$, then $R=\{(a,b)\}$, being a subset of $A\times B$ is a relation from $A$ to $B$. For that matter, given any two sets $A$ and $B$, the empty relation $\emptyset \subseteq A\times B$ is always a relation from $A$ to $B$. So not only can relations consists of just one single pair, they can also consists of no pairs at all.

If this seems useless to you, and in a sense we hardly ever really care about such simple relations, then you are, in a sense, correct. However, just because something is not particularly complicated or interesting does not mean we should discard it. For instance, you may argue that the empty set is kinda useless. True, we will never study it since there is not much we can say. but it is useful, for instance to express that two sets are disjoins by saying $A\cap B=\emptyset$. Categorically and notationally, it would be a disaster to discard of it since it will force upon you very cumbersome formulations of results. For roughly the same reasons we like to have all relations, even trivial-looking ones.

Related Question