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Definition of a relation

  1. Jan 15, 2009 #1
    1. The problem statement, all variables and given/known data
    This is a seemingly subtle point here, that would actually clear up both of the two previous posts I have made. A relation R is said to be defined on S and T if [tex] s \in S [/tex] and [tex] s \in dom(R) [/tex].

    2. Relevant equations

    3. The attempt at a solution
    Does this mean, that if I see a question that starts if R is defined on S ... that I can assume if I define a relation on S, call it T, that the domain of T must also be S. Or for any relation that we define on a set, it can be assumed that the domain of the relation is that set?
  2. jcsd
  3. Jan 15, 2009 #2


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    Hi icantadd! :smile:

    Sorry, I'm not following any of that. :confused:

    A relation on S is a subset of S x S.

    From the PF Library page on relation …
  4. Jan 15, 2009 #3


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    A relation on S is any subset of the cartesian product SxS, the set of ordered pairs of objects from S. It does not follow from that that every member of S must be in some ordered pair. For example, I could define R on Z, the set of integers by "xRy is x and y are both odd numbers" That would consist of things like (1, 1), (3, 5), (-3, 7), etc. That is also of course, a relation on "O", the set of odd integers. If R is a relation on both sets S and T, the members of the pairs of R must be contained in both S and T: some subset of the intersection of S and T.
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