for some integers with .
ℤ/3ℤ
Show that for all classes:
_ _
x,y ∈ ℤ/3ℤ
The following is true:
_ _ ______ ______
∀x1, x2 ∈ x, y1,y2 ∈ y : x1 + y1 = x2 + y2
I dont now how to make lines directly above the letters
Okay so I dont know if I need that, but what does the ℤ/3ℤ say? Am I creating classes depending on Euclidean division?
divisible by 3: 0:= [0] = {...,-9,-6,-3,0,3,6,9,...}
remainder 1 1:= [1] = {...,-8,-5,-2,1,4,7,10,13,...}
remainder 2 2:= [2] = {...,-10,-7,-4,-1,2,5,8,11,14,...}
How can I combine it with the statement I have to prove?
Additionally I need to show that (ℤ/3ℤ , +) is an Abelian group.
In this case I I would draw a matrix and analyze it:
+ 0 1 2
0 0 1 2
1 1 2 0
2 2 0 1
Okay so 0 doesnt change anything. This means that the neutral element is 0
The invers is 0, because it is a part of every row.
It is also symmetrical on the diagonal line.
This means it is an Abelian group.
Is this proof enough?
Oh okay, it just looks so complicated to me, that I thought it is an actual solution. Sadly, my point stands still. I dont know what to do with it. What are the qs?
Sorry for being such a dumbo. But I seriously dont know how to approach this problem.
As I said, I used the division algorithm. So, means is the quotient when is divided by 3 and is the remainder. Just go back to the Division Algorithm, and it should all make sense.
Now, to show that , you need to show that is divisible by 3.