Please note that this is a problem from a Book of Abstract Algebra by Pinter. Now, in his book, Pinter refers to ℤ as the additive group of integers, which he alternatively denotes as <ℤ,+>. Also, Pinter refers to ℝ as the additive group of real numbers, which he alternatively denotes as <ℝ,+>.
Now, I've made the following observations so far:
1) If we add any rational number, which is in ℝ, to ℤ, then we will get a class of equivalent rational numbers. (A) Now, if we do this for every rational number in ℝ and take the union of all of these disjoint classes, then we will obtain a set containing all of the rational numbers. Furthermore, (B) if we apply this same reasoning to adding every irrational number in ℝ to ℤ, then we will obtain a set that contains all of the irrational numbers.
2) Let {ℤ+a:a is in ℚ} represent the set described in (A) and let {ℤ+b: b is in ℝ-ℚ} be the set described in (B). Now, these two sets would be disjoint, and if we take the union of these two sets, then they would produce the set of real numbers.
With that said, this was just brainstorming on my part, so I would appreciate any feedback on accurately describing the cosets in this question. Thanks for your time and attention.
Best Answer
The equivalence class (i.e. coset) of any number $x \in \Bbb R$ will be made of the translates of $x$ by integer numbers, i.e. it will be $x + \Bbb Z = \{x + z \mid z \in \Bbb Z\}$. You now need to look for a representative for every such coset. Since every $x \in \Bbb R$ may be written as $x = [x] + \{x\}$ with $[x]$ being the integer part and $\{x\} \in [0,1)$ the fractional part, and since two real numbers have the same coset if and only if they differ by an integer, if and only if they have the same fractional part, it follows that each coset $\hat x \in \Bbb R / \Bbb Z$ may be identified with its fractional part $\{x\} \in [0,1)$. Conversely, for every $q \in [0,1)$ we have a coset in $\Bbb R / \Bbb Z$, namely the coset of $q$, i.e. $q + \Bbb Z$.
The group law on $[0,1)$, that I shall denote as "$\oplus$" in order for you not to mistake it for the usual "$+$", is given by $q \oplus r = \{q + r \}$, the fractional part of $q+r$. It can be proven that the quotient group $\Bbb R / \Bbb Z$ is isomorphic to the group $\big( [0,1), \oplus \big)$.