# Can this notation be used to describe relations that are not functions?

I have only seen the notation $f : A \rightarrow B$ used for discussing functions. Can this notation be used for relations that are not functions?

For example, $\tan : \mathbb R \rightarrow \mathbb R$ appears to define a relation that is not left-total (for example $\frac \pi 2\in \mathbb R$ is not mapped to anything in codomain $\mathbb R$) and hence not a function. Would it be incorrect to express this relation, that is not a function, using this notation?

Edit: To borrow a great example from Arthur's answer below, consider the notation $\le\; : \mathbb R \rightarrow \mathbb R$. This seems to me to unambiguously express a relation that, while left-total is certainly not functional (e.g. $(3, 3), (3, 4) \in \;\le$) and hence not a function. If it's unambiguous, surely this notation can be used to express this relation that is not a function...

I guess I could express my question more clearly as: is there anything in the expression $f:A\rightarrow B$ that asserts that $f$ is not merely a relation but specifically a function?

## 2 Answers

There is a category of relations having sets as objects and binary relations as arrows.

In that context the notation: $$R:A\to B$$ where $R\subseteq A\times B$, is fine.

In my view (somewhat coloured by categories) the expression $f:A\to B$ on its own is not enough to state that we are dealing with a function (or a relation).

I do recognize it as an arrow in a category and the context must make clear which category.

So my answer to your last question is: "no".

An entirely correct way would be $\tan\subseteq \Bbb R\times\Bbb R$. I think $\tan: \Bbb R\to\Bbb R$ would be considered technically incorrect but in many circumstances within allowable abuse of notation, mostly because $\tan$ is still a partial function defined on most of the given domain. Writing something like $\leq{}:\Bbb R\to\Bbb R$ is probably taking it too far.

• Rather then "function-like" would "Partial function" be more useful? Aug 24 '18 at 7:21
• @QthePlatypus You're right. I fixed it up. Aug 24 '18 at 7:22
• Hmm. I think the $\tan$ you are referring to (when you say it is a subset of $\mathbb R \times \mathbb R$) is not just function-like, but a function in all senses, because $\frac \pi 2$, for example, is (usually implicitly) not part of the domain. In my example I explicitly said that the domain was the entire $\mathbb R$... My example is not the relation we generally understand by $\tan$ stated on its own. Aug 24 '18 at 7:23