Results 1 to 12 of 12

Math Help - Basic relation question

  1. #1
    Member
    Joined
    Mar 2010
    Posts
    96

    Basic relation question

    One quick question about relations.

    I know how is defined a relation,
    but how is defined R^2 and R^n?

    If we have a relation R = {(1,2),(2,3),(3,4)}.

    How do we get a R^2 and, R^n. If n is let say 504, surely we won't go calculating R * R * R... 504 times.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,663
    Thanks
    1616
    Awards
    1
    Quote Originally Posted by Nforce View Post
    If we have a relation R = {(1,2),(2,3),(3,4)}.
    How do we get a R^2 and, R^n. If n is let say 504, surely we won't go calculating R * R * R... 504 times.
    Can you explain what "calculating R * R * R... 504 times" means.
    What does R*R even mean in this context?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Mar 2010
    Posts
    96
    R is a relation. We can calculate with relations. In school we have defined R^-1 is an inverse relation, the product of two relations is R * S, and so on.
    R^2 means. R times R. The product of two same relations.

    I was hoping that somebody would explain to me the product of two relations and R^n.

    on this example R = {(1,2),(2,3),(3,4)}.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Nov 2010
    From
    Staten Island, NY
    Posts
    451
    Thanks
    2
    If A is a set, then a relation R on A is a subset of A^n for some n. In this case R is an n-ary relation.

    The relation you gave is a 2-ary or binary relation.

    I don't think there is a standard definition for multiplying relations. So you will need to give us the definition you are using.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,663
    Thanks
    1616
    Awards
    1
    Quote Originally Posted by Nforce View Post
    R is a relation. We can calculate with relations. In school we have defined R^-1 is an inverse relation, the product of two relations is R * S, and so on.
    R^2 means. R times R. The product of two same relations.
    I was hoping that somebody would explain to me the product of two relations and R^n.
    on this example R = {(1,2),(2,3),(3,4)}.
    I sorry to tell you, but that reply makes no sense whatsoever.
    After teaching this material for forty+ years, I have never seen a product of a relation defined.
    It is true that for \mathcal{R}=\{(1,2),(2,3),(3,4)\} we can have composition of relations.

     \mathcal{R}\circ\mathcal{R}=\{(1,3),(2,4)\} and
     \mathcal{R}\circ(\mathcal{R}\circ\mathcal{R})=\{(1  ,4)\}
    After that the compositions are empty.

    Is that what you are asking?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Mar 2010
    Posts
    96
    I didn't know that there's not a standard definition for calculating with relations. Hmm why is that?



    The first is definition for a inverse relation. And second is for a product of relation R and S.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Senior Member
    Joined
    Nov 2010
    From
    Staten Island, NY
    Posts
    451
    Thanks
    2
    Quote Originally Posted by Nforce View Post
    I didn't know that there's not a standard definition for calculating with relations. Hmm why is that?



    The first is definition for a inverse relation. And second is for a product of relation R and S.
    OK - now these definitions make sense. So by product you mean compositiion. Plato has then given you R^n for each natural number n in his post above.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,663
    Thanks
    1616
    Awards
    1
    Quote Originally Posted by Nforce View Post
    Does your text book really define relation composition as:
    \mathcal{R}\circ\mathcal{S}=\{(x,z):\exists y,(x,z)\in \mathcal{R} \wedge (z,y)\in \mathcal{S}\}~???
    If so I would say it is at least fifty years behind current practice.

    The answer I gave you is based on this defition:
    \mathcal{R}\circ\mathcal{S}=\{(x,z):\exists y, (x,z)\in \mathcal{S} \wedge (z,y)\in \mathcal{R}\}
    That is, relation composition is like function composition right to left, inside out.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Member
    Joined
    Mar 2010
    Posts
    96
    I still don't know if we are talking about the same thing.
    I have found an example in the book. Would you be so kind to explain me. We always said the product of relations.



    The inverse relation is obvious. But the product as we defined it is (how I see it). The first element of order pair and the second element from another order pair is written in a new order pair. And so on...
    Attached Thumbnails Attached Thumbnails Basic relation question-relation2.jpg  
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,537
    Thanks
    778
    Composition and product are sometimes the same thing, for example, when linear operators in a vector space are represented as matrices. Also, application of a mapping F to an argument x is often written just as F x, which resembles a product. For this reason, I guess, some textbooks may refer to composition of relations as a product, but the correct term is composition.

    Also, the standard it to write the relation that is applied first on the right and the one applied second on the left. As Plato said, this works smoothly when relations are functions since (f\circ g)(x) stands for f(g(x)). Using Internet lingo, there is a temptation to think that authors who don't follow this convention are trolling to create confusion, though this is unlikely, of course.

    Other than that, the definition of composition you have is the standard one.
    Follow Math Help Forum on Facebook and Google+

  11. #11
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,663
    Thanks
    1616
    Awards
    1
    Quote Originally Posted by emakarov View Post
    Other than that, the definition of composition you have is the standard one.
    Do you seriously think that? Standard?
    So far as I know, that has not been in use since the 1940’s.
    Even Quine changed his notation from 1938 to 1951.
    Can you give us a reference to a modern text using a ‘left to right’ definition for relation compositions?
    I do have an algebra text by Marie Weiss at Tulane in the ’40 & 50’s that does use that notation. She was interested in permutation groups so that makes a certain amount of sense (as a product).
    Follow Math Help Forum on Facebook and Google+

  12. #12
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,537
    Thanks
    778
    Other than that, the definition of composition you have is the standard one.
    I meant, other than the order (left/right). I guess, now it looks like it is I who is trolling to create confusion.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: September 21st 2011, 07:21 PM
  2. [SOLVED] Relation Question 2
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: September 4th 2011, 06:57 AM
  3. Relation Question
    Posted in the Discrete Math Forum
    Replies: 13
    Last Post: September 3rd 2011, 07:28 PM
  4. Question about relation and set
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: October 31st 2008, 06:47 AM
  5. relation question
    Posted in the Algebra Forum
    Replies: 1
    Last Post: September 17th 2008, 04:57 PM

Search Tags


/mathhelpforum @mathhelpforum