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?