Discrete mathematics, defining (+, *) operations on a equivalence relations.

Question

In a set: $\mathbb N/xSy$

($xSy$ defined on natural numbers and as $k|(x - y)$ and $k$ is a natural number)

We define: $+_k, *_k$ as:

$$[x]_S +_k[y]_S = [x + y]_S$$ $$[x]_S *_k[y]_S = [xy]_S$$

Check for what k these definitions are correct.

I have no idea how to do this.

Answer 1

See Well-definedness under an equivalence relation : the result of the operation defined on the equivalence classes is independent from the choice of the "representatives" of the equivalent classes.

This means that the opeartion (e.g. $+$) must satisfies the property :

if $x_1 S x_2$ and $y_1 S y_2$ then $(x_1+y_1)S(x_2+y_2)$.

Thus, the problem asks to chek for what values of $\text k$ the above holds.

---

Example

Consider $k=2$.

We have that $7S5$ because $2 | (7-5)$.

Thus : $5,7 \in [5]_S$.

And $4,12 \in [4]_S$, because $2 | (12-4)$.

Consider now : $[4]_S +_k [5]_S = [4+5=9]_S$.

If we "change representatives", we have e.g. : $[12]_S +_k [7]_S = [12+7=19]_S$ and $2 | (19-9)$.

So, the definition of $+_k$ seems to work for $k=2$.

For a proof, we have to use the property :

if $x_1 S x_2$ and $y_1 S y_2$ then $(x_1+y_1)S(x_2+y_2)$.

Thus, we have that $k | (x_1-x_2)$ and $k | (y_1-y_2)$.

This means :

$x_1=kn +x_2$ and $y_1=km +y_2$.

Thus :

$x_1+y_1=kn +x_2 + km +y_2 = k(n+m) + (x_2+y_2)$,

i.e.

$k | ((x_1+y_1)-(x_2+y_2))$.