Results 1 to 14 of 14

Thread: Relation Question

  1. #1
    Member
    Joined
    Jul 2011
    Posts
    196

    Relation Question

    Let $\displaystyle R$ be a relation on $\displaystyle \mathbb{Z}$, where $\displaystyle xRy \Leftrightarrow x|9y$. State whether R is reflexive, symmetric, transitive or an equivalence relation.

    I'm having trouble understand the notation | in the question. I am used to this meaning "such that". The text doesn't explain this problem very well; it says that x must = 9ky were $\displaystyle k \in \mathbb{Z}$

    How do I go about determining the relation here?

    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,782
    Thanks
    2824
    Awards
    1

    Re: Relation Question

    Quote Originally Posted by terrorsquid View Post
    Let $\displaystyle R$ be a relation on $\displaystyle \mathbb{Z}$, where $\displaystyle xRy \Leftrightarrow x|9y$. State whether R is reflexive, symmetric, transitive or an equivalence relation.
    I'm having trouble understand the notation | in the question. I am used to this meaning "such that". The text doesn't explain this problem very well; it says that x must = 9ky were $\displaystyle k \in \mathbb{Z}$
    The notation $\displaystyle a|b$ usually means that $\displaystyle a\text{ divides }b$.
    But that does really make sense here. The textbook should have defined it somewhere in the text.

    If it does mean divides the the relation cannot be reflexive because 0 does not divide 0.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Nov 2010
    From
    Staten Island, NY
    Posts
    451
    Thanks
    2

    Re: Relation Question

    It probably means "divides." It's not symmetric since 3R2, but not 2R3.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Jul 2011
    Posts
    196

    Re: Relation Question

    The answer given is:

    R is reflexive since $\displaystyle x|9x$

    R is not symmetric since if x = 2 and y = 18, then $\displaystyle 2|162$ but 2 is not divisible by 18.

    R is not transitive since if x = 9y and y = 9z then x = 81z, which clearly does no divide 9z. (I don't understand why 81z doesn't divide 9z? is it something to do with the fact that R is is on $\displaystyle \mathbb{Z}$ ?)
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Jul 2011
    Posts
    196

    Re: Relation Question

    So, knowing that | means divides, what is the best method to determine the nature of the relationship? Just to take arbitrary integers and try to find a contradiction?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,163
    Thanks
    46

    Re: Relation Question

    Quote Originally Posted by terrorsquid View Post
    R is reflexive since $\displaystyle x|9x$
    The reason is that for every $\displaystyle x\in\mathbb{Z}$ we have $\displaystyle 9x=9\cdot 1\cdot x$ .
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Senior Member
    Joined
    Nov 2010
    From
    Staten Island, NY
    Posts
    451
    Thanks
    2

    Re: Relation Question

    More simply for transitive: Use x=81, y=9, z=1. Then xRy (because 81|81), yRz (because 9|9), but not xRz (because 81 does not divide 9).
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Senior Member
    Joined
    Nov 2010
    From
    Staten Island, NY
    Posts
    451
    Thanks
    2

    Re: Relation Question

    Hmmm... is 0|0 true? I think the definition of "divides" is a|b if there is an integer k such that b=ka. Under this definition 0|0 is true since 0=k*0 for ANY integer k. Can someone with a better memory than me remind me if k must be unique.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,163
    Thanks
    46

    Re: Relation Question

    Quote Originally Posted by DrSteve View Post
    Hmmm... is 0|0 true?
    Yes , $\displaystyle 0= 1\cdot 0$ i.e. there exists $\displaystyle k\in\mathbb{Z}$ such that $\displaystyle 0=k\cdot 0$ .
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Member
    Joined
    Jul 2011
    Posts
    196

    Re: Relation Question

    Quote Originally Posted by terrorsquid View Post
    The answer given is:

    R is not symmetric since if x = 2 and y = 18, then $\displaystyle 2|162$ but 2 is not divisible by 18.
    With the answer given here, I understand the first test 2 divides 9*(18)... but why is the second test 18 divides 2? Shouldn't it be x = 18 and y = 2 and $\displaystyle 18 | 9(2)$ which is true?
    Follow Math Help Forum on Facebook and Google+

  11. #11
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,163
    Thanks
    46

    Re: Relation Question

    Quote Originally Posted by terrorsquid View Post
    I'm having trouble understand the notation | in the question. I am used to this meaning "such that". The text doesn't explain this problem very well; it says that x must = 9ky were $\displaystyle k \in \mathbb{Z}$
    This problem is creating confusion. Taking into account that $\displaystyle a|b$ is an universal agreement for $\displaystyle a$ divides $\displaystyle b$, Did you mean $\displaystyle 9y=kx$ where $\displaystyle k\in\mathbb{Z}$?. Could you transcribe the exact formulation?
    Follow Math Help Forum on Facebook and Google+

  12. #12
    MHF Contributor

    Joined
    Aug 2006
    Posts
    21,782
    Thanks
    2824
    Awards
    1

    Re: Relation Question

    Quote Originally Posted by FernandoRevilla View Post
    This problem is creating confusion. Taking into account that $\displaystyle a|b$ is an universal agreement for $\displaystyle a$ divides $\displaystyle b$, Did you mean $\displaystyle 9y=kx$ where $\displaystyle k\in\mathbb{Z}$?. Could you transcribe the exact formulation?
    On this webpage is the following quote:
    "Clearly, $\displaystyle 1|n$ and $\displaystyle n|n $. By convention, $\displaystyle n|0$ for every $\displaystyle n$ except 0 (Hardy and Wright 1979, p. 1)"
    Follow Math Help Forum on Facebook and Google+

  13. #13
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,163
    Thanks
    46

    Re: Relation Question

    Quote Originally Posted by Plato View Post
    "Clearly, $\displaystyle 1|n$ and $\displaystyle n|n $. By convention, $\displaystyle n|0$ for every $\displaystyle n$ except 0 (Hardy and Wright 1979, p. 1)"
    Let us see. Conventions are conventions. If we define in $\displaystyle \mathbb{Z}$ $\displaystyle x|y$ iff $\displaystyle y=kx$ for some $\displaystyle k\in\mathbb{Z}$ then, $\displaystyle |$ is reflexive (no problem). If we exclude $\displaystyle 0|0$ then $\displaystyle |$ is not reflexive by convention (no problem). That depends on the context: in the context of Measure Theory $\displaystyle 0\cdot \infty =0$ (by convention), in the context of infinite products $\displaystyle 1^{\infty}=1$ (by convention) etc. For those reasons and looking at all the previous answers I suggested to terrorsquid to transcribe the exact formulation of the problem, taking (besides) into account that he,or the problem (who knows?) says $\displaystyle x|y\Leftrightarrow x=9ky$ for some $\displaystyle k\in\mathbb{Z}$ .
    Last edited by FernandoRevilla; Sep 3rd 2011 at 10:55 AM.
    Follow Math Help Forum on Facebook and Google+

  14. #14
    Member
    Joined
    Jul 2011
    Posts
    196

    Re: Relation Question

    I wrote the question down word for word in my original post. The next part I mentioned was from the explanation of the solution - not the question itself. I wont be able to reference the exact problem again as the questions are randomly generated in our online text quiz section with unique solution explanations. I could reference another similar question from the same section; it will depend on which question the quiz generates but they are always the same type of question just with slightly different values.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Relation Question 2
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: Sep 4th 2011, 06:57 AM
  2. Generators and relation question
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: May 21st 2010, 06:25 AM
  3. Equivalence Relation question
    Posted in the Discrete Math Forum
    Replies: 5
    Last Post: Sep 18th 2009, 04:06 AM
  4. Question about relation and set
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: Oct 31st 2008, 06:47 AM
  5. relation question
    Posted in the Algebra Forum
    Replies: 1
    Last Post: Sep 17th 2008, 04:57 PM

Search Tags


/mathhelpforum @mathhelpforum