Hello,

I have three relations on the set of natural numbers defined like this.

1) $\displaystyle x$ is in relation with $\displaystyle y$, when $\displaystyle x+y$ can be divided by $\displaystyle k$.

2)$\displaystyle x$ is in relation with $\displaystyle y$, when $\displaystyle x-y$ can be divided by $\displaystyle k$.

3)$\displaystyle x$ is in relation with $\displaystyle y$, when $\displaystyle x-y$ is equal to $\displaystyle k$.

I need to find $\displaystyle k$ such that $\displaystyle k$ is a positive number and when put into those relations they are equivalent.

So for the first one I think that $\displaystyle k$ can only be either 1 or 2, because it has to be reflexive, and any number is divisible by 1, and any sum of two same numbers is divisible by two.

For the second one k can be any number except 0 because every number can divide 0, which we get when we test the reflexiveness of the relation, and for any number $\displaystyle n$ we can find $\displaystyle m$ such that m is a sum of any multiple of k and $\displaystyle k-$$\displaystyle n$ MOD $\displaystyle k$, and then $\displaystyle n+m$ is divisible by k.

In the third one the only possible value of k is 0, because the relation has to be reflexive and when we substract two of the same numbers we get a 0. When $\displaystyle k$ is 0 the relation is also symetrical and transitive.

Is this correct?