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If you define a relation R on N = {0,1,2,3, ... }
What are the equivalence classes? Is it 8 of them from like [0]...[7] ?
Last edited by LuisRodg (2008-02-23 05:16:49)
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The equivalence classes are the classifications of squares modulo 8.
And
So squares of integers can only be ≡ 0, 1 or 4 (mod 8). There are therefore 3 equivalence classes: {odd integers (whose squares ≡ 1 (mod 8)}, {even integers divisible by 4 (whose squares ≡ 0 (mod 8))}, {even integers not divisible by 4 (whose squares ≡ 4 (mod 8))}.
Last edited by JaneFairfax (2008-02-23 06:31:44)
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Pretty cool problem, this is one of the more "interesting" equivalence classes.
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
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Jane, I do not understand what your doing there. My professor didnt teach us anything like that to find the equivalence classes, she just said "plug numbers in and you will get them" which doesnt really help at all. I like having a definite way of finding them like you showed up there but I dont understand it?
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What Ive shown is that given any integer n,
.An integer k is in the same equivalence class as n if and only if
. Hence, to find all the different equivalence classes to which k can belong, you find all possible values of .Offline
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