Relation vs Function: Understanding N-ary Relations

[Mr Davis 97]

My professor informally defined a n-ary relation as a "function" that assigns to an n-tuple from arbitrary sets $X_1, X_2, ... X_n$ a well-formed statement that either holds or does not hold. I know that this definition is somewhat informal, but how can the professor use the word function if functions themselves are defined in terms of relations?

 

[FactChecker]

If your professor was informally defining a relation and you understood what he meant, then he accomplished his goal. If he was trying to make be formal, then you have a point and he should have used different words. Proper formal definitions are often very obscure.

 

[Stephen Tashi]

My professor informally defined a n-ary relation as a "function" that assigns to an n-tuple from arbitrary sets $X_1, X_2, ... X_n$ a well-formed statement that either holds or does not hold.

That is one way of saying that a set has a precise definition. For a set S to be well defined, for each "X" there must be a rule $R(X)$ that determines whether $X \in S$ or $X \not \in S$. That rule can be regarded as a function from the set of whatever $X$ may come from to the set of truth values {True, False}. So your professor's statement is technically correct.

However, the usual way of phrasing it would be simply to say that an n-ary relation is a :"set of n-tuples" take (respectively) from some sets $X_1,X_2,...X_n$. When something is called a "set" in mathematics it is automatically taken to mean a well defined set.

but how can the professor use the word function if functions themselves are defined in terms of relations?

Mathematics is seldom presented in a strict and orderly way such that each concept uses only concepts defined previously. People who study the foundations of mathematics in a very detailed way are interested in ways that mathematics can developed and defined in a strict order. However, in other branches of mathematics, the concepts likes functions and sets are taken for granted and not developed "from scratch" in a strict order. The attention to order of presentation is reserved for the more advanced material of the course - for example, limits have to be presented in order to define derivatives.