The relation $\sim$ in $\mathbb{R}$ is defined as: $x \sim y \iff x − y \in \mathbb{Z}$

Question

$(a) \quad$ Show that $\sim$ is an equivalent relation.

$(b) \quad$ Give $2$ distinct equivalent classes (must show they are distinct).

$(c) \quad$ $[0, 1) = \{x \in \mathbb{R} \mid 0 \leq x < 1\}.$ Show that $^{\mathbb{R}}\big/_{\sim} \cong [0, 1).$

I was able to prove that the relation is reflexive, symmetric and transitive. So, it is equivalent.

For $(b)$ one class I could find was

$$\{y \in \mathbb{R} \mid x \sim y\} = \{y \in \mathbb{R} \mid \exists z \in \mathbb{Z}, x - y = z\} = \{y \in \mathbb{R} \mid \exists z \in \mathbb{Z}, x = z + y\}.$$

But I cannot find another one which is distinct!

And for $(c)$ I don't even know how to start this proof. Any ideas? Thanks

Answer 1

(a) It's called equivalence relation

(b) The set you provided here is the equivalence class of a fixed element $x$. For example what is the equivalence class of $0$, or of $1/2$?

(c) Consider a specific function $f:\Bbb R\to [0,1)$ which satisfies $f(x)=f(y)\iff x\sim y$.

Answer 2

In your answer for b) you have to declare what $x$ is.

For instance if $x = \pi$ then we can define the equivalence class of pi is

$\{y\in \mathbb R: \pi\sim y\} = \{y\in \mathbb R|\exists z \in \mathbb Z; \pi-y = z\}=\{y\in \mathbb R| y=-z-\pi;z\in \mathbb Z\} = \{k-\pi|k\in \mathbb Z\}=$

$\{..., -3-\pi, -2-\pi, -1, -\pi, 1-\pi, 2-\pi, 3-\pi, ....\}$.

You can do this for any real number $x$.

Do it for $x = \sqrt 2$ and get $\{y\in \mathbb R: \sqrt 2 \sim y\} = \{k-\sqrt 2|k\in \mathbb Z\} = \{...., -1 -\sqrt 2, -\sqrt 2, 1-\sqrt 2, 2-\sqrt 2,....\}$

Or eve for $x = 3$. The equivalence class for $3$ is $\{k-3|k\in \mathbb Z\} = \mathbb Z$.

To show $\{..., -3-\pi, -2-\pi, -1, -\pi, 1-\pi, 2-\pi, 3-\pi, ....\}$ and $\{...., -1 -\sqrt 2, -\sqrt 2, 1-\sqrt 2, 2-\sqrt 2,....\}$ are distinct, is a matter of saying

Let $x \in \{y\in \mathbb R: \sqrt 2 \sim y\}\cap \{y\in \mathbb R: \pi\sim y\}$. let $\pi \sim x$ and $\sqrt 2 \sim x$. So $\pi -x = k\in \mathbb Z$ and $\sqrt 2-x= m\in \mathbb Z$ for some $k,m$. But that means $x= \pi - k = \sqrt 2-m$ and that means $\pi = \sqrt 2 +(k -m)$ or that $\pi \sim 2$ (which follows by symmetry and transitivity).

And that's nonsense. SO the two classes are distinct.