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    If aRb <=> ab is a perfect square and is a equivalence relation, I'm not sure how I can show its equivalence classes. Can somebody help me please.
    I've already shown that it is an equivalence relation but can't be bothered to put it all up.
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    Equivalence classes are not fun!
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    (Original post by boromir9111)
    Equivalence classes are not fun!
    Same! I want to get this exam over with! Even though we'll be doing slightly different stuff . How do you find the module? I find this one really boooooooring.
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    (Original post by JBKProductions)
    I'm not sure how I can show its equivalence classes. Can somebody help me please.
    Are you just struggling to write them clearly or are you struggling to identify them?
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    (Original post by Get me off the £\?%!^@ computer)
    Are you just struggling to write them clearly or are you struggling to identify them?
    Hey! I'm struggling to identify them and then maybe write them when i understand it .
    Thanks for your reply.
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    (Original post by JBKProductions)
    Hey! I'm struggling to identify them and then maybe write them when i understand it .
    Thanks for your reply.
    OK, so try writng some down.

    I'll get you started {2,8,18,32,50.....}
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    (Original post by Get me off the £\?%!^@ computer)
    OK, so try writng some down.

    I'll get you started {2,8,18,32,50.....}
    hmmm...I'm not sure what you mean. I don't understand how you got that set.
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    (Original post by JBKProductions)
    hmmm...I'm not sure what you mean. I don't understand how you got that set.
    Take any pair a,b from that set and you have a~b.

    e.g.
    2~8 since 2*8=4^2
    8~18 since 8*18=12^2
    2~18 since 2*18=6^2
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    (Original post by Get me off the £\?%!^@ computer)
    Take any pair a,b from that set and you have a~b.

    e.g.
    2~8 since 2*8=4^2
    8~18 since 8*18=12^2
    2~18 since 2*18=6^2
    Oh I see. I have the answer here as {mn^2: n is an element of N}. But it's still really confusing how they got that.
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    (Original post by JBKProductions)
    Oh I see. I have the answer here as {mn^2: n is an element of N}. But it's still really confusing how they got that.
    The set I wrote is the case where m=2.

    Did they not have anything to say about m?
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    (Original post by Get me off the £\?%!^@ computer)
    The set I wrote is the case where m=2.

    Did they not have anything to say about m?
    They said this: "for some fixed square-free number m (i.e., m is a product of distinct
    prime numbers, so that m is not divisible by the square of any number greater than 1)." But I just don't seem to understand what they mean. I've read it like 10 times!
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    Well if you write out a few of the classes explicitly you then only need to find a neat way to describe them (the answer they gave).

    {1,4,9,16,25.......}
    {2,8,18,32,50.....}
    {3,12,27,48,75...}
    {5,20,45,80,125...}
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    (Original post by Get me off the £\?%!^@ computer)
    Well if you write out a few of the classes explicitly you then only need to find a neat way to describe them (the answer they gave).

    {1,4,9,16,25.......}
    {2,8,18,32,50.....}
    {3,12,27,48,75...}
    {5,20,45,80,125...}
    Oh ok. I sort of understand it now, but is there a reason why you didn't write the set {4, 16, 36, 64, 100,...}?
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    (Original post by JBKProductions)
    Oh ok. I sort of understand it now, but is there a reason why you didn't write the set {4, 16, 36, 64, 100,...}?
    because all those numbers are equivalent to 1.

    the equivalence classes partition the set. In this case the equivalence classes are

    [1], [2], [3], [5], [6], [7], [10], [11], [13], ...

    where [m] denotes the equivalence class of m.
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    (Original post by RichE)
    because all those numbers are equivalent to 1.

    the equivalence classes partition the set. In this case the equivalence classes are

    [1], [2], [3], [5], [6], [7], [10], [11], [13], ...

    where [m] denotes the equivalence class of m.
    May sound like a silly question but how is it equivalent to 1? what do you mean by that?
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    (Original post by RichE)
    because all those numbers are equivalent to 1.

    the equivalence classes partition the set. In this case the equivalence classes are

    [1], [2], [3], [5], [6], [7], [10], [11], [13], ...

    where [m] denotes the equivalence class of m.
    Oh I see. Is it because those numbers are contained within the equivalence class [1]? Like {1, 4, 9, 16...} and [4] = {4, 16,...}?
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    (Original post by boromir9111)
    May sound like a silly question but how is it equivalent to 1? what do you mean by that?
    It means that 4R1 and 16R1 and so on, and so {4, 16, 36, 64, 100,...} isn't the whole equivalence class, since it also contains 1, 9, 25, 49, 81, 121, etc... .

    The equivalence class of a number a \in \mathbb{N} is the set \{b\, :\, ab \text{ is a perfect square} \}, denoted by [a]. Note that if b \in [a] then [a]=[b].

    So [1] = {1, 4, 9, 16, 25, ...}, since 1×each of these numbers is a perfect square. Similarly for any square number, [n²] = {1, 4, 9, 16, 25, ...}, since if a=m² and b=n² then ab=(mn)² is a perfect square, so a and b must be in the same equivalence class.

    Similarly, [2] = {2, 8, 18, ...} is the set of numbers which when doubled give a perfect square. That is, it's the set of "even squares halved"... and so on; and [13] = {13, 52, ...} is the set of squares who have 13 as a divisor divided by 13... etc.

    EDIT:
    (Original post by JBKProductions)
    Oh I see. Is it because those numbers are contained within the equivalence class [1]? Like {1, 4, 9, 16...} and [4] = {4, 16,...}?
    No. 4~1, so [4]=[1], and so [4]={1, 4, 9, 25, 36, ...} as well. How could the equivalence classes of two equivalent numbers be different?
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    (Original post by JBKProductions)
    Oh I see. Is it because those numbers are contained within the equivalence class [1]? Like {1, 4, 9, 16...} and [4] = {4, 16,...}?
    You wrote [4]={4,16,...}

    Are you clear that [1]=[4]=[9]=etc = {1,4,9,16,25.....}?
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    Oh ok I see. I understand it now. I think. so [1] = [16]?
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    (Original post by JBKProductions)
    Oh ok I see. I understand it now. I think. so [1] = [16]?
    Yup.
 
 
 
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