What is the Equivalence Class for the given Equivalence Relation?

[arnold28]

Homework Statement

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

Homework Equations

??

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 don't know how i start to build it when i have |x| = |y|
its probably something very easy and i just don't get it for some reason

 

[George Jones]

Suppose x is given but unknown, and that |x| = |y|. What can y equal in terms of the given x?

 

[arnold28]

hmmm...y must always be +x or -x?
but i don't 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?

 

[George Jones]

Now think about concrete examples. If x = -3, what can y be? Consequently, what is [-3]?

 

[arnold28]

if x = -3, then y can then be 3 or -3
What is [-3]? I don't know, {..., ?, -3, ?, ...}

 

[George Jones]

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|}

 

[HallsofIvy]

if x = -3, then y can then be 3 or -3
What is [-3]? I don't 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 set[\b] of all numbers equivalent to -3 so [3]= ?

 

[arnold28]

I thought I understood the b) part, but now I am 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 don't think I understood the theory deeply enough =)

 

[George Jones]

I thought I understood the b) part, but now I am 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.

can [-3] then be only {-3,3} in the a) -part?
And [-2] = {-2,2} etc?

Yes.

I'm confused because we only had those modulus examples in the class and in book and I don't think I understood the theory deeply enough =)

It looks like you are catching on.

What about [0] in a)?

 

[arnold28]

[0] must then be only {0}

What about R: R <-> R, where xRy, iff floor(x) = floor(y)

i don't know if floor() is the right way to write floor function, but can't 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 a lot!

 

[George Jones]

[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.

What about R: R <-> R, where xRy, iff floor(x) = floor(y)

i don't know if floor() is the right way to write floor function, but can't 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.

 

[Gigasoft]

http://en.wikipedia.org/wiki/Interval_(mathematics)