Prove $\lbrace A_n : n \in \mathbb{Z}\rbrace$ is a partition of $\mathbb{Q}$

Question

Dr. Pinter's "A Book of Abstract Algebra" presents the following exercise:

Prove that each of the following is a partition of the indicated set. Then describe the equivalence relation with that partition.

For each integer $n$, let $A_n = \lbrace x \in \mathbb{Q} : n \leqslant x < n + 1\rbrace$. Prove $\lbrace A_n : n \in \mathbb{Z}\rbrace$ is a partition of $\mathbb{Q}$.

Here's the picture that I have of $A_n, \mathbb{Q}, \text{ and } \mathbb{Z}$:

Please comment on my picture, as well as guide me on how to complete this proof.

Answer 1

Your picture shows $A_n$ as a subset of $\Bbb Z$, which is not true for any $n$: the only integer in $A_n$ is the integer $n$ itself. The rest of $A_n$ consists of the rational numbers between $n$ and $n+1$.

Here’s a better picture:

 --)[-------)[-------)[-------)[-------)[-------)[-------)[-------)[-----  
   -2       -1        0        1        2        3        4        5

the line represents $\Bbb Q$, and the interval shown as

                                    [-------)  
                                    n      n+1

represents $A_n$.