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Thread: 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, $\displaystyle \sin 0 = \sin \pm\pi =\sin \pm2\pi = \dots$. So $\displaystyle 0, \pm\pi, \pm2\pi, \dots$ will all be in the same equivalence class.

    In general, if $\displaystyle \sin x = \sin y$, then $\displaystyle x = n\pi +(-1)^ny$. 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 $\displaystyle (x,y)$ and $\displaystyle (u,v)$ related:

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

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

    (e) Think of some numbers $\displaystyle x$, $\displaystyle y$ for which $\displaystyle \lfloor x\rfloor = $
    $\displaystyle \lfloor y\rfloor$. E.g. $\displaystyle \lfloor 5\rfloor = \lfloor 5.1\rfloor = \lfloor 5.99\rfloor = \lfloor 5.99999\rfloor$. 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: $\displaystyle x/R = \left[ x \right] = \left[ {\left\lfloor x \right\rfloor ,\left\lfloor x \right\rfloor + 1} \right)$.

    For part c: define a mapping $\displaystyle f:\mathbb{R} \mapsto \left[ {0,2\pi } \right]\text{ as }f(x) = \left[ {x - \left\lfloor {\frac{x}
    {{2\pi }}} \right\rfloor \left( {2\pi } \right)} \right]$.
    Now we can define the classes.
    $\displaystyle x/R = \left[ x \right] = \left[ {f(x)} \right]$.
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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: $\displaystyle x/R = \left[ x \right] = \left[ {\left\lfloor x \right\rfloor ,\left\lfloor x \right\rfloor + 1} \right)$.

    ...$\displaystyle x/R = \left[ x \right] = \left[ {f(x)} \right]$.
    ... 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 $\displaystyle /$ is being used in two different ways when a relation $\displaystyle R$ is defined on a set $\displaystyle S$:

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

    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 $\displaystyle [x]$ or $\displaystyle [x]_R$ 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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