Math Is Fun Forum

LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

Equivalence classes on a relation.

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)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Equivalence classes on a relation.

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)

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Equivalence classes on a relation.

Pretty cool problem, this is one of the more "interesting" equivalence classes.

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"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..."

LuisRodg

Real Member

Registered: 2007-10-23

Posts: 322

Re: Equivalence classes on a relation.

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?

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Equivalence classes on a relation.

What I’ve 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

.