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Further

i) Give a geometric description of the equivalence class to which(0,0) belongs.

ii) Is the number of equivalence classes finite?

My approach is as below:

Equivalence Relation

Reflexive: $\forall (x,y) \in \mathbb{R}^2, (x,y)\sim(x,y)$ since $x-x=0\in \mathbb{Z}$

Symmetric: $(x_1,y_1)\sim(x_2,y_2)\Rightarrow x_1-x_2\in \mathbb{Z}$. Now since $x_2-x_1=-(x_1-x_2)\in \mathbb{Z}$ we have $(x_2,y_2)\sim(x_1,y_1)$

Transitive: ($(x_1,y_1)\sim(x_2,y_2)$ and $(x_2,y_2)\sim(x_3,y_3)) \Rightarrow$ ($x_1-x_2\in \mathbb{Z}$ and $x_2-x_3\in \mathbb{Z}$). Now since sum of any two integers is again an integer we have $(x_1-x_2)+(x_2-x_3)=(x_1-x_3)\in \mathbb{Z}$ And so $(x_1,y_1)\sim(x_3,y_3)$

Geometric Description

We know that the equivalence class to which $(0,0)$ belongs is equal to $[(0,0)]$. Now $0-x \in \mathbb{Z} \Rightarrow x \in \mathbb{Z}$. Hence $[(0,0)]=\{(x,y) \in \mathbb{R}^2 \mid x \in \mathbb{Z}\}$.

Geometrically ~ contains every line through integer values of $x$ and parallel to $y$ axis.

Number of Equivalence classes

This is where I am stuck. How do I determine whether the number of classes is finite or infinite?

Please let me know if I have made any mistake in my approach. Also please help with finite/infinite aspect of the number of classes

Bernard
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Rajkumar Kumawat
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    Pick $x_1, x_2\in [0;1)$ with $x_1\neq x_2$, then $[(x_1;0)]\neq [(x_2;0)]$ (prove this). Therefore, we have uncountably many different equivalence classes. – Severin Schraven Feb 20 '20 at 08:27
  • @SeverinSchraven In this case we have $[(x_1,0)]={(x_1+k,y)\mid k\in \mathbb{Z}}$ and $[(x_2,0)]={(x_2+k,y)\mid k\in \mathbb{Z}}$. And $(x_1+k_1,y)\neq (x_2+k_2,y)$ as per our choice of $x_1\neq x_2$. Please Let me know if this reasoning is correct. – Rajkumar Kumawat Feb 20 '20 at 09:11
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    You also have to use that $x_1, x_2 \in [0;1)$, $x_1 \neq x_2$ is not enough (as for example $x_1= 0.1$ and $x_2=1.1$ doesn't work despite $0.1\neq 1.1$). – Severin Schraven Feb 20 '20 at 16:04
  • @SeverinSchraven Got it. Thanks – Rajkumar Kumawat Feb 20 '20 at 16:20

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