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A binary relation $\approx$ is defined on $\mathbb R$ as follows: $$\forall x,y\in \Bbb R, x\approx y \iff \lfloor x \rfloor = \lfloor y \rfloor$$

As for finding all values of $x\in \Bbb R$ that satisfy $x^2 \in [36]$, I understand it is the set of integers. Would there be any special cases that do not fulfill the following relation?

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  • $\begingroup$ I edited my answer, please read it $\endgroup$
    – Astyx
    Commented Oct 30, 2016 at 12:17

1 Answer 1

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Hint Let $n$ be an integer. On what condition does $\lfloor x\rfloor = n$ ?

Answer : We have $\lfloor x\rfloor = n \iff x\in[n; n+1[$

Since $\lfloor x\rfloor$ always is an integer what follows is that the equivalence classes are $E_n = \{x\in \Bbb R, x\in[n; n+1[\}$ for $n \in\Bbb Z$

Thus $[36] = E_{36} = \{x\in \Bbb R, x\in[36; 37[\}$.

Therefore $$\begin{align} x^2\in[36] &\iff 36\le x^2 \lt37\\ &\iff x \in ]-\sqrt{37}; -6]\cup [6; \sqrt{37}[ \end{align}$$

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  • $\begingroup$ Exactly, now what can you deduce from this ? $\endgroup$
    – Astyx
    Commented Oct 30, 2016 at 10:51
  • $\begingroup$ Since $\lfloor x\rfloor$ is always an integer, the equivalence classes are $E_n = \{x\in\Bbb R, \lfloor x \rfloor = n\}$ for $n\in\Bbb Z$ $\endgroup$
    – Astyx
    Commented Oct 30, 2016 at 11:23
  • $\begingroup$ The set of equivalence classes is countable and you can select a representative of each equivalence class as an integer. $\endgroup$
    – IamThat
    Commented Oct 30, 2016 at 12:09