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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.

Kernel Sohcahtoa
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  • What you've said is true but it doesn't answer the question. To do that, you need to understand intuitively and describe what an arbitrary coset is. – Qudit Mar 23 '17 at 03:52
  • If $x \mod 1 \equiv y \mod 1$, what can you say about $x$ and $y$? – erfink Mar 23 '17 at 03:54
  • Just a minor point about notation: wouldn't you rather have "$\langle \mathbb Z, + \rangle$" instead of "<ℤ,+>"? I know \langle \mathbb Z, + \rangle is a lot more typing, but it's a lot less hunting for Unicode characters. – Robert Soupe Mar 23 '17 at 04:27
  • I usually think of $\mathbb{R}$ as "time" with the subgroup $\mathbb{Z}$ marking off the start of each day. Then an example of a coset is just all the points representing "8:45 am"; and so on. – ancient mathematician Mar 23 '17 at 08:52
  • @erfink. xmod1 has a remainder of zero and ymod1 has a remainder of zero. So if xmod1≡ymod1, then can we say that x and y are congruent modulo 1 or x≡ymod(1) since x and y have the same remainder when divided by one? I'm not sure where to go with the suggestion that you provided. Any further help would be appreciated. Thanks – Kernel Sohcahtoa Mar 23 '17 at 15:15
  • @Qudit. I appreciate your answer and feedback. However, I'm afraid that I'm unable to intuitively understand and describe what an arbitrary coset is. Could you expand a bit further? Thank you for your time and attention. – Kernel Sohcahtoa Mar 23 '17 at 15:21
  • @ancient mathematician. Based on my understanding, if G is a group and H is a subgroup of G, then the cosets of H form a partition of G. So each coset is like an equivalence class. So, as per your example, if we think of R as time, then all the points representing midnight would be one equivalence class? – Kernel Sohcahtoa Mar 23 '17 at 15:26
  • Yes, and all the two o'clock in the morning another, etc. An old-fashioned analogue clock is a physical manifestation of the quotient, time goes round and round if we ignore the days! – ancient mathematician Mar 23 '17 at 15:33

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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 $\big( [0,1), \oplus \big)$.

Alex M.
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