In the additive group $\mathbb{Q}/\mathbb{Z}$ prove that there is only one representative q for all cosets $0\le q<1$

Question

I have clearly understood the fact that the cosets would be of the form q+$\mathbb{Z}$ where $q$ lies between $0$ and $1$ and $\mathbb Z$ will be the kernel of some homomorphism but what I didn't get is that why would there be only one representation for the coset?can someone help me to understand the intuition..

Answer 1

If $q_1$ and $q_2$ belong to the same coset and $0\leqslant q_1<1$, then $q_2-q_1\in\mathbb Z$. So, if we also have $0\leqslant q_2<1$, then $q_2-q_1\in(-1,1)$. This, together with the fac that $q_2-q_1\in\mathbb Z$, implies that $q_2-q_1=0$, since $(-1,1)\cap\mathbb Z=\{0\}$.