Is there a fundamental mathematical function that requires 3 inputs or more?

Question

So a mathematical operation can be represented as a function that maps inputs to outputs. For example "sin(x)" is a function that maps 1 input to 1 output, and "a + b" maps 2 inputs to 1 output. My question is is there a function that requires a minimum of 3 inputs on a fundamental level? I'm not talking about something like "a + b + c" even though that has 3 inputs because the steps of solving that function are evaluated 2 variables at a time ie "(a + b) + c". Is there something that needs 3 or more inputs to make sense in the same way addition needs 2? I don't have very strong math background so I didn't know how to google my question properly or how to tag it here properly. Thank you for your patience.

Answer 1

Here's a class of examples you've probably seen before: we can take constructions from Euclidean geometry and view them as functions. For example, we have the functions $OC$, $IC$, $CC$ which take three points and return respectively the orthocenter, incenter, and circumcenter of the triangle they form (fine, with an appropriate convention in case they don't actually form a triangle).

There isn't really a good way to view these as being built out of binary functions in a precise sense: there is no continuous function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ such that for all points $x,y,z$ in the plane, the orthocenter (or whatever) of the triangle formed by $x,y,z$ is determined just by $x$ and $f(y,z)$. This is a bit tricky to prove, but I think it shows reasonably convincingly that $OC/IC/CC$ are "fundamentally ternary" operations.

EDIT: As JMoravitz points out below, there's no real reason to single out "degenerate triangles" (e.g. two vertices equal). This may seem confusing at first, and if you're not familiar with being comfortable with "degenerate objects" the following is a good exercise: can you find the right definition of the (say) orthocenter of a "triangle" with two vertices the same?

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That said, there are two ways we can still think of them as being built out of binary operations:

• We can use Currying, as in mr_e_man's comment. This always works, but the drawback to it is that the new simpler functions we produce have much more complicated ranges (functions of points, functions of functions of points, etc.) so - while mathematically very important! - it feels a bit like cheating.

• We can also drop the requirement of continuity that I mentioned. If we allow any binary function $f$ to be a "building block" for our ternary functions, then we'll always be able to succeed if our domain is infinite: just pick your favorite bijection between the domain squared and the domain (in this case, a bijection between $\mathbb{R}^2$ and $\mathbb{R}$). These functions, however, are generally quite pathological and again feel like cheating.

Answer 2

Thanks for your answers and comments! A conditional statement is to me a satisfactory answer. In order for a conditional statement to make sense it requires (a) the condition, (b) the yield if the condition evaluates to true, and (c) the yield if the condition evaluates to false. To the best of my knowledge all three components are required. Sorry if this answer is not satisfactory to your interpretation of the question.