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Math Help - very confused since I missed class...relations

  1. #1
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    very confused since I missed class...relations

    Describe the partition for each of the following equivalence relations:

    (c) for x, y belong to R, xRy iff sinx=siny
    (d) For (x,y) and (u,v) belong to RxR (x,y)S(u,v) iff xy=uv=0 or xyuv>0
    (e) For x,y belong to R, xTy iff [x]=[y] where [x] is defined to be the greatest integer in x.

    Thanks for help on any of them!
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    Equivalence Classes

    Hello zhupolongjoe
    Quote Originally Posted by zhupolongjoe View Post
    Describe the partition for each of the following equivalence relations:

    (c) for x, y belong to R, xRy iff sinx=siny
    (d) For (x,y) and (u,v) belong to RxR (x,y)S(u,v) iff xy=uv=0 or xyuv>0
    (e) For x,y belong to R, xTy iff [x]=[y] where [x] is defined to be the greatest integer in x.

    Thanks for help on any of them!
    (c) Any given equivalence class will contain numbers whose sines are equal. For example, \sin 0 = \sin \pm\pi =\sin \pm2\pi = \dots. So 0, \pm\pi, \pm2\pi, \dots will all be in the same equivalence class.

    In general, if \sin x = \sin y, then x = n\pi +(-1)^ny. So all the numbers in a given equivalence class will be related in this way.

    (d) Think about the rules you are given that make (x,y) and (u,v) related:

    • Either xy = uv = 0. In other words, at least one of x or y and at least one of u and v must be zero.
    • Or xyuv > 0. In other words, the product of all four numbers x, y, u and v is positive. So what does this tell you about the individual signs of x, y, u and v?

    Then use the fact that whenever two ordered pairs (x,y) and (u, v) are related by one or other of these rules, then they will be in the same equivalence class.

    (e) Think of some numbers x, y for which \lfloor x\rfloor =
    \lfloor y\rfloor. E.g. \lfloor 5\rfloor = \lfloor 5.1\rfloor = \lfloor 5.99\rfloor = \lfloor 5.99999\rfloor. So all these numbers will be in the same equivalence class.

    Grandad
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    You've been extremely helpful to me today. I really appreciate it. Thank you!
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    This post is simply to expand on the previous post.
    Here are the representations of equivalence classes.

    For part e: x/R = \left[ x \right] = \left[ {\left\lfloor x \right\rfloor ,\left\lfloor x \right\rfloor  + 1} \right).

    For part c: define a mapping f:\mathbb{R} \mapsto \left[ {0,2\pi } \right]\text{ as }f(x) = \left[ {x - \left\lfloor {\frac{x}<br />
{{2\pi }}} \right\rfloor \left( {2\pi } \right)} \right].
    Now we can define the classes.
    x/R = \left[ x \right] = \left[ {f(x)} \right].
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    Equivalence Class: Notation

    Hello everyone -

    With reference to my previous criticism of the use of the 'quotient set' / notation (http://www.mathhelpforum.com/math-he...tml#post285435), I see that Plato is using it here ...
    Quote Originally Posted by Plato View Post
    ...For part e: x/R = \left[ x \right] = \left[ {\left\lfloor x \right\rfloor ,\left\lfloor x \right\rfloor  + 1} \right).

    ... x/R = \left[ x \right] = \left[ {f(x)} \right].
    ... to represent an equivalence class. This is a use of the notation with which I am not familiar. However, if it is accepted these days, then I withdraw my previous criticism.

    So you need to understand that / is being used in two different ways when a relation R is defined on a set S:

    • S/R represents the quotient set, or the set of all the equivalence classes.
    • If x \in S, x/R represents the single equivalence class that contains the element x. Other notations that are (I think!) in more common use to represent this class are [x] and [x]_R.

    Perhaps Plato would like to comment on this?

    Grandad
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    Quote Originally Posted by Grandad View Post
    Perhaps Plato would like to comment on this?
    I just assumed that the two notations are in common use.
    Equivalence relation - Wikipedia, the free encyclopedia

    P.S. Both notations are found in the 1969 ed. of NAIVE SET THEORY by Paul Halmos.
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    Use of '/' notation

    Quote Originally Posted by Plato View Post
    I just assumed that the two notations are in common use.
    Equivalence relation - Wikipedia, the free encyclopedia

    P.S. Both notations are found in the 1969 ed. of NAIVE SET THEORY by Paul Halmos.
    I have scoured many internet articles (including the Wikipedia one you quote) that mention the '/' notation and, although I have found many references to 'set / relation' to represent the quotient set, I have yet to find one that uses 'element / relation' to represent the equivalence class. Not that that proves it's not used in this way, of course.

    But personally, I would steer clear of it, and stick to the widely accepted [x] or [x]_R to represent the equivalence class. In my experience, students often find relations confusing enough without using / to mean two different things!

    Grandad
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