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  1. #1
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    Relation

    Hello. I need help on determining this relation.
    Determine the relation \rho defined with x\rho y \Leftrightarrow (\exists z)(y=zx) for Z numbers
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    Quote Originally Posted by javax View Post
    Hello. I need help on determining this relation.
    Determine the relation \rho defined with x\rho y \Leftrightarrow (\exists z)(y=zx) for Z numbers
    what do you mean "determine the relation"? isn't it already determined. it is given.

    note that, geometrically speaking, the relation describes the set of all straight lines that pass through the origin with integer slope (i suppose you mean \mathbb{Z} when you wrote Z).
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    Quote Originally Posted by Jhevon View Post
    what do you mean "determine the relation"? isn't it already determined. it is given.

    note that, geometrically speaking, the relation describes the set of all straight lines that pass through the origin with integer slope (i suppose you mean \mathbb{Z} when you wrote Z).
    Sorry for my english, I haven't had the chance to see these terms in english, I guess. Yes I mean \mathbb{Z}
    I mean determine if it is equivalence relation or order relation. That's what it is required!

    Thanks
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    Quote Originally Posted by javax View Post
    Sorry for my english, I haven't had the chance to see these terms in english, I guess. Yes I mean \mathbb{Z}
    I mean determine if it is equivalence relation or order relation. That's what it is required!

    Thanks
    it is not an equivalence relation. it is an (partial) order relation.

    can you see why?
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  5. #5
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    Quote Originally Posted by Jhevon View Post
    it is not an equivalence relation. it is an (partial) order relation.

    can you see why?
    Nope, I need that prove. C'mon help me
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    Quote Originally Posted by javax View Post
    Nope, I need that prove. C'mon help me
    i want to help you develop one yourself.

    here's how to start, what ae the definitions of "equivalence relation" and "(partial) order relations"? how do we know a relation is one of these or not?
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    Quote Originally Posted by Jhevon View Post
    i want to help you develop one yourself.

    here's how to start, what ae the definitions of "equivalence relation" and "(partial) order relations"? how do we know a relation is one of these or not?
    Here we go:
    1. (r) (\forall a\in A) a\rho a
    2. (s) (\forall a,b \in A) a\rho b \Rightarrow b\rho a
    3. (t) (\forall a,b,c \in A) a\rho b \wedge b\rho c \Rightarrow  a\rho c
    4. (a) (\forall a,b\in A)a\rho b \wedge b\rho a \Rightarrow a = b

    The relation that fulfills conditions 1, 2 & 3 is called equivalence relation. Relation that fulfills 1, 3 & 4 is called a partial order relation. That's what I know
    Last edited by javax; January 22nd 2009 at 01:59 PM.
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    Quote Originally Posted by javax View Post
    Here we go:
    1. (r) (\forall a\in A) a\rho a
    2. (s) (\forall a,b \in A) a\rho b \Rightarrow b\rho a
    3. (t) (\forall a,b,c \in A) a\rho b \wedge b\rho c \Rightarrow  a\rho c
    4. (a) (\forall a,b\in A)a\rho b \wedge b\rho a \Rightarrow a = b

    The relation that fulfills conditions 1, 2 & 3 is called equivalence relation. Relation that fulfills 1, 2 & 4 is called a partial order relation. That's what I know
    actually, 1, 3 & 4 makes a partial order relation.

    ok, so check if it is an equivalence relation:

    (1) does x relate to itself? can you find and integer such that x = zx?
    (2) do we have symmetry. if we can find and integer z so that y = zx, will we always be able to find an integer n so that x = ny?
    (3) do we have transitivity? if we have integers n and m so that a = mb and b = nc, can we find an integer k so that a = kc?
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  9. #9
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    Quote Originally Posted by Jhevon View Post
    actually, 1, 3 & 4 makes a partial order relation.

    ok, so check if it is an equivalence relation:

    (1) does x relate to itself? can you find and integer such that x = zx?
    (2) do we have symmetry. if we can find and integer z so that y = zx, will we always be able to find an integer n so that x = ny?
    (3) do we have transitivity? if we have integers n and m so that a = mb and b = nc, can we find an integer k so that a = kc?
    After exploring my notebook, here's my attempt:

    (1) (r)
    x\rho x \Leftrightarrow \exists k=1 such that x=kx \Rightarrow x=x
    it fills the reflexive condition

    (2) (s) x\rho y (\exists z)(y=zx), \hspace{2cm}y\rho x (\exists u)(x=uy)
    if we multiply these we have (\exists z)(\exists u) yx=z x u y For z=u=1 we have yx=xy (Does this mean that we have a symmetry?

    (3) (t)  \exists z(y=zx)\wedge \exists u(t=uy). I multiplied again
    (\exists u)(\exists z)  t=(uz)x.
    sub. uz=p \Rightarrow t=px.

    What do you think?
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by javax View Post
    After exploring my notebook, here's my attempt:

    (1) (r)
    x\rho x \Leftrightarrow \exists k=1 such that x=kx \Rightarrow x=x
    it fills the reflexive condition
    brilliant!

    (2) (s) x\rho y (\exists z)(y=zx), \hspace{2cm}y\rho x (\exists u)(x=uy)
    if we multiply these we have (\exists z)(\exists u) yx=z x u y For z=u=1 we have yx=xy (Does this mean that we have a symmetry?
    think of an example.

    take y = 2 and x = 1. clearly y relates to x, since y = zx (where z = 2).

    but does x relate to y? can we find an integer k so that x = ky, that is, 1 = k*2?


    (3) (t)  \exists z(y=zx)\wedge \exists u(t=uy). I multiplied again
    (\exists u)(\exists z)  t=(uz)x.
    sub. uz=p \Rightarrow t=px.

    What do you think?
    yup, that's nice. we do have transitivity
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  11. #11
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    Quote Originally Posted by Jhevon View Post

    think of an example.

    take y = 2 and x = 1. clearly y relates to x, since y = zx (where z = 2).

    but does x relate to y? can we find an integer k so that x = ky
    y=zx, x=ky
    Ok like you said if we have y=x=z=k=1, we have a symmetry don't we?
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  12. #12
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by javax View Post
    y=zx, x=ky
    Ok like you said if we have y=x=z=k=1, we have a symmetry don't we?
    i gave you an example where it didn't work, didn't i? look back at your definitions, it must work for ALL x and y, not just specific ones
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  13. #13
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    Quote Originally Posted by Jhevon View Post
    i gave you an example where it didn't work, didn't i? look back at your definitions, it must work for ALL x and y, not just specific ones
    ohhhh cool
    ok so if it is not symmetric does that mean that it is antisymmetric?
    Thanks for your time
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  14. #14
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    Quote Originally Posted by javax View Post
    ohhhh cool
    ok so if it is not symmetric does that mean that it is antisymmetric?
    Thanks for your time
    no, anti-symmetric is defined by the fourth definition you have. after reading a few math books you should start to see how mathematicians define things. if anti-symmetric means "not symmetric" that is exactly how they would define it. it shouldn't be hard to come up with an example of a relation that is both not symmetric and not anti-symmetric. see here

    you have to check if it is anti-symmetric. if it is, we have an order relation (since you already showed we have reflexivity and transitivity)
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