# Relation vs Function: Understanding N-ary Relations

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• Mr Davis 97
In summary, the conversation discusses the informal definition of a n-ary relation as a "function" that assigns a well-formed statement to an n-tuple from arbitrary sets. While this definition may be technically correct, it is not the standard way of phrasing it, as a n-ary relation is typically described as a set of n-tuples. The use of the word function in this definition may seem contradictory, as functions themselves are defined in terms of relations. However, in mathematics, concepts such as functions and sets are often taken for granted and not strictly developed in a particular order.
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?

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.

Mr Davis 97 said:
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.

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FactChecker

## 1. What is the difference between a relation and a function?

A relation is a connection or association between two or more elements, whereas a function is a specific type of relation where each input has only one corresponding output. In other words, a function is a special type of relation that follows the one-to-one or many-to-one rule, while a relation can be one-to-one, one-to-many, many-to-one, or many-to-many.

## 2. How can n-ary relations be represented?

N-ary relations can be represented in several ways, including tables, graphs, matrices, and set-builder notation. In tables, n-ary relations are represented by listing all the elements in columns and rows, with the connections between the elements indicated in the corresponding cells. In graphs, n-ary relations are represented by nodes (or vertices) connected by edges (or arcs). Matrices use rows and columns to represent the elements and their connections, while set-builder notation uses set brackets, a colon, and a condition to represent the elements and their relationships.

## 3. What is the importance of understanding n-ary relations?

Understanding n-ary relations is essential in many fields, including mathematics, computer science, and linguistics. In mathematics, n-ary relations are used to model complex systems and analyze data. In computer science, n-ary relations are used in databases, knowledge representation, and artificial intelligence. In linguistics, n-ary relations are used to study language and its structure.

## 4. How do you determine if a relation is a function?

To determine if a relation is a function, you can use the vertical line test. If a vertical line intersects the graph of the relation at more than one point, then the relation is not a function. Another way to determine if a relation is a function is by checking if each input has only one corresponding output. If there is more than one output for an input, then the relation is not a function.

## 5. Can a relation be both a function and a non-function?

No, a relation cannot be both a function and a non-function. A relation is either a function or not a function, depending on the number of outputs for each input. However, a relation can be a function for some inputs and not a function for others, depending on its domain and range. For example, a relation can be a function for positive numbers but not for negative numbers.

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