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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. :tongue:

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Equivalence classes are not fun!
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 :tongue:. How do you find the module? I find this one really boooooooring.
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?
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 :colondollar:.
Thanks for your reply.
Original post by JBKProductions
Hey! I'm struggling to identify them and then maybe write them when i understand it :colondollar:.
Thanks for your reply.


OK, so try writng some down.

I'll get you started {2,8,18,32,50.....}
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.
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
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. :s-smilie:
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. :s-smilie:


The set I wrote is the case where m=2.

Did they not have anything to say about m?
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! :frown:
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...}
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,...}?
Reply 13
Avatar for RichE
RichE
15
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.
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?
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,...}?
Reply 16
Avatar for nuodai
nuodai
17
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∈Na \in \mathbb{N} is the set {b : ab is a perfect square}\{b\, :\, ab \text{ is a perfect square} \}, denoted by [a][a]. Note that if b∈[a]b \in [a] then [a]=[b][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?
(edited 13 years ago)
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.....}?
Oh ok I see. I understand it now. I think. so [1] = [16]?
Original post by JBKProductions
Oh ok I see. I understand it now. I think. so [1] = [16]?


Yup. :smile:

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