I'm trying to solve an exercise from my math lecture. The question is:
Let $k \in \mathbb{Z}$, $(G, \circ) := (\mathbb{Z}_{12}, \oplus)$, $(H, *) := (\mathbb{Z}_{8}, \oplus)$ and $\varphi: G \ni [a]_{12} \mapsto [6a + k]_8 \in H$.
Prove that $\varphi$ is well defined.
Based on another exercise which maps $[a]_{12} \mapsto [6a]_8$, I've tried to do this like this:
$$[a]_{12} = [b]_{12} \\ \iff a \equiv b\pmod {12} \\ \iff 12 | b-a \\ \iff (\exists l \in \mathbb{Z})[12l = b-a]$$
Now insert a new variable, $k$.
$$ \iff (\exists l,k \in \mathbb{Z})[12l = (b + k) -(a+k)] \\ \iff (\exists l,k \in \mathbb{Z})[72l = 6(b+k)-6(a+k)] \\ \iff (\exists l,k \in \mathbb{Z})[8l = 6(b+k)-6(a+k)] \\ \iff 8 | (6b+k) - (6a+k) \\ \iff 6a + k \equiv 6b+ k \pmod{8} \\ \iff [6a +k]_8$$
My problem is actually the introducing $k$ part. This does not look right. I think I can simplify $72l$ to $8l$ because 72 is a multiple of 8, but how should I properly introduce $k$ here?
Also, looked at from a set theorey side, if I see $\varphi$ as a relation, is it correct that the set would be like this?
$$\mathcal{R} = \{([0]_{12}, [0+k]_8), ([0]_{12}, [1+k]_8), \ldots\} $$
i.e. that the $k$ is treated as a constant in the set?
From what I can tell from your definition of $\varphi$, $k$ is indeed a "constant", in the sense that you need to prove that for any fixed $k \in \mathbb{Z}$ you can define a map $\varphi_k$.
So it is not an "existence of $k$" matter! You fix $k$ in the beginning, and then argue as $$[a]_{12} = [b]_{12} \\ \iff a \equiv b\pmod {12} \\ \iff 12 | b-a \\ \iff (\exists l \in \mathbb{Z})[12l = b-a] 12l = b-a \\ 72l = 6b-6a \\ $$ so, for any fixed $k \in \mathbb{Z}$ $$72l = 6b-6a +k - k \\ \text{ so }\exists m \in \mathbb{Z} \text{ such that } 8m = (6b+k)-(6a+k) \\ \text{ so } 8 | (6b+k) - (6a+k) \\ \text{ so } 6a + k \equiv 6b+ k \pmod{8} \\ $$ and you are done.