Ok, so I want to show that there are 3 eq classes for eq relation xRy 3|x+2y

so I have [0] = {x in Z: 2x, x = 3k} [3] = {x in Z: 2x+3, x = 3k} [x} = {x in Z: x=x}

but I'm not sure if I'm writing this correctly

so your relation is x is equivalent to y iff 3 divides x+2y correct?

yes

So, i don't want the answer given to me by the way

just need to understand what i'm doin

\[[0]=\{y\in \mathbb{Z}:0\sim y\}\] \[[0]=\{y\in \mathbb{Z}: 3|0+2y\}=\{y\in \mathbb{Z}: 3|2y\}=\{y\in \mathbb{Z}: y\text{ is a multiple of 3}\}\]

so is that how you write the proof of the eq class?

\[[1]=\{y\in \mathbb{Z}: 3|1+2y\}=\{y\in \mathbb{Z}: 1+2y=3k\}=\{\cdots, -2,1,4,7,\cdots\}\]

what are you trying to prove?

He's writing out carefully for you the equivalence classes, and has given you two of them so far. If the question is what are all of the equivalence classes, the next question is: what other classes are there beyond these two, if any?

ok, i already figured out the [0] and [1], but I wrote them differently

but i want to show that if x and y are the same, that it works too

remember that the equivalence classes form a partition of the underlying set.

Notice that you wrote above in your original statement that [3] is an equivalence class different from [0]; that is not the case.

oh i see

so could i write:

[n] = {y in Z: n + 3n = 3k} to show that when x and y are the same it's another class or does that not count?

no

that is wrong on a couple levels

there are only 3 EQ classes...I gave you two of them

note that 1~1 and 2~2 but that doesn't mean that they are in the same EQ class. this is just a consequence of the relation being reflexive.

ok so I think the other class is [3}

[3]

no

don't tell me

[0]=[3]

it must be [2]

correct

so i would say y is in z: 2 + 2y = 3k

k is a multiple of 2

k is some integer

it will have to be true that it will be a multiple of 2....but i would still say k is some integer

because the remainder is 2?

right, so it's even multiples of 3

you could also write out the set in roster notation

6, 12, 18

2k+2

?

\[\{\ldots,-1,2,5,8,\ldots\}\]

sigh

K+3

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