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I Relation vs. Function

  1. Jan 20, 2017 #1
    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?
     
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  3. Jan 21, 2017 #2

    FactChecker

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    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.
     
  4. Jan 21, 2017 #3

    Stephen Tashi

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    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.

    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.
     
    Last edited: Jan 21, 2017
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