# Finding Equivalence Class

by arnold28
Tags: class, equivalence
 P: 14 1. The problem statement, all variables and given/known data Find the equivalence class [2] for the following equivalence relations: a) R: Z <-> Z, where xRy, iff |x| = |y| b) T: N <-> N, where xTy, iff xmod4 = ymod4 N means natural numbers etc...there wasnt the correct symbols in the latex reference 2. Relevant equations ?? 3. The attempt at a solution Ok so I know how to do the b) part, because we had examples at the class, its: [0] = {0,4,8,12,...} [1] = {1,5,9,13,...} [2] = {2,6,10,14,...} so the answer is [2] = {2,6,10,14,...} right? but i dont know how i start to build it when i have |x| = |y| its probably something very easy and i just dont get it for some reason
 Mentor P: 6,248 Suppose x is given but unknown, and that |x| = |y|. What can y equal in terms of the given x?
 P: 14 hmmm...y must always be +x or -x? but i dont understand how the classes are formed. For example class [0], does it mean the list starts at 0? In the b-part the list increases always by 4, but what about in this, by 1?
 Mentor P: 6,248 Finding Equivalence Class Now think about concrete examples. If x = -3, what can y be? Consequently, what is [-3]?
 P: 14 if x = -3, then y can then be 3 or -3 What is [-3]? I dont know, {..., ???, -3, ???, ...}
 Mentor P: 6,248 Do you understand why the answer to b) is the answer to b)? Back to a). [x] = {y | xRy} = {y in Z | |y| = |x|} [-3] = {y | (-3)Ry} = {y in Z | |y| = |-3|}
Math
Emeritus
Thanks
PF Gold
P: 39,683
 Quote by arnold28 if x = -3, then y can then be 3 or -3 What is [-3]? I dont know, {..., ???, -3, ???, ...}
Yes, if |x|= |y| and x= -3, then y can be either 3 or -3. So the only numbers equivalent to -3 are 3 and -3. [-3] is the [b]set[\b] of all numbers equivalent to -3 so [3]= ?
 P: 14 I thought I understood the b) part, but now im not sure if i do deeply enough. So, in each class the elements are "equivalent" in the way the equivalence relation is defined? xmod4 = ymod4 means every element which has same modulus when divided by 4 belong to same class? can [-3] then be only {-3,3} in the a) -part? And [-2] = {-2,2} etc? I'm confused because we only had those modulus examples in the class and in book and I dont think I understood the theory deeply enough =)
Mentor
P: 6,248
 Quote by arnold28 I thought I understood the b) part, but now im not sure if i do deeply enough. So, in each class the elements are "equivalent" in the way the equivalence relation is defined? xmod4 = ymod4 means every element which has same modulus when divided by 4 belong to same class?
Yes.
 Quote by arnold28 can [-3] then be only {-3,3} in the a) -part? And [-2] = {-2,2} etc?
Yes.
 Quote by arnold28 I'm confused because we only had those modulus examples in the class and in book and I dont think I understood the theory deeply enough =)
It looks like you are catching on.

 P: 14 [0] must then be only {0} What about R: R <-> R, where xRy, iff floor(x) = floor(y) i dont know if floor() is the right way to write floor function, but cant find the correct symbol. [2] is then something like {2, 2.1, 2.2, ... , 2.99999...} but what is the correct way to write it? Because 2 can have any amount of decimals after it. Does it have to be in a list form like a) and b) here was? Thanks much for the replies, you helped me alot!
Mentor
P: 6,248
Quote by arnold28
[0] must then be only {0}[/qu0te]

This shows that different equivalence classes for the same equivalence relation don't have to have the same number of elements, i.e., in a), [-3] has two elements and [0] has one element.

 Quote by arnold28 What about R: R <-> R, where xRy, iff floor(x) = floor(y)
i dont know if floor() is the right way to write floor function, but cant find the correct symbol. [2] is then something like {2, 2.1, 2.2, ... , 2.99999...} but what is the correct way to write it? Because 2 can have any amount of decimals after it. Does it have to be in a list form like a) and b) here was?
No, it doesn't have to be a list. For example, you can specify [2] by using inequalities.
 P: 59

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