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Definition of a relation

  1. Jul 3, 2012 #1
    It is kinda strange. There is no agreement on the definition of a relation.
    Some books says it is a set of ordered pairs.
    Other books says it is a subset of a cartesian product.
    How nice if everything can be agreed down to a few axioms like Euclid's elements.

    What is your favourite definition of a relation?
  2. jcsd
  3. Jul 3, 2012 #2
    The two definitions say the same thing.
  4. Jul 3, 2012 #3
    I do not agree with that. In the definition using ordered pairs, it is assumed that a set can be built from ordered pairs. But in the cartesian definition, a set is provided already, you just use a part of it via subset.
  5. Jul 7, 2012 #4
    I go with:

    [tex]R \mbox{ is a relation} \Leftrightarrow \forall r(r \in R \Rightarrow \exists x \exists y (r=<x,y>))[/tex]

    This definition states that the elements of R are ordered pairs if and only if R is a relation.
    Last edited: Jul 7, 2012
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