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some relation $R$ is defined on $\mathbb{R}$ such that $xRy \ \iff \ x = 7^{k}y,$ for some $k\in \mathbb{Z}$. Prove that $R$ is an equivalence relation

I'm confused with proving that it is symmetric and transitive.

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  • $\begingroup$ For symmetric, note that $k$ can be negative. $\endgroup$ – wjm Jul 8 '19 at 13:56
  • $\begingroup$ For transitive, note that $7^a7^b=7^{a+b}$. $\endgroup$ – wjm Jul 8 '19 at 13:57
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If $x\sim y$ and $y\sim z$ then we have $x=7^ky$ and $y=7^nz$ so $$x = 7^k(7^nz) = 7^{k+n}z$$ so it is transitive.

Also $x\sim y$ then we have $x=7^ky$ so $y =7^{-k}x$ so it is also a symmetric.

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Symmetric: If $x\sim y$ then there exists an integer $k$ so that $x=7^ky$ Then $y=7^{-k}x$ So $y\sim x$.

Transitive: Suppose $x\sim y$ and $y\sim z$ so that there exist integers $k,m$ such that $x=7^ky$ and $y=7^mz$, from which it follows that $x=7^{k+m}z$. Thus $x\sim z$

Reflexive: For every $x$, we have $x\sim x$ because $x=7^0x$.

So this is an equivalence relation.

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This holds not only for $\,G = 7^{\large \Bbb Z}\,$ but also for any set with the same algebraic (group) structure (closed under products & inverses) and the proof is more conceptual (and just as simple) done this way.

Suppose that $\,G\subset \Bbb R\,$ satisfies $\,\color{#0a0}1\in G\,$ and $\,g,h\in G\,\Rightarrow\, \color{#08f}{gh}\in G,\,$ and $\, \color{#c00}{g^{-1}}\in G$

Then $\ x\approx y \!\overset{\rm def}\iff\! x = g\, y\ \ {\rm for\ some}\ \ g\in G\ $ is an equivalence relation, $ $ since

$\qquad\ \ \ \ \approx\,$ is $\ \ \rm\color{#0a0}{reflexive}\,\ \ $ by $\ x = \color{#0a0}1x\,\Rightarrow\, x\approx x$

$\qquad\ \ \ \ \approx\,$ is $\,\rm\color{#c00}{symmetric}\,$ by $\ x\approx y\,\Rightarrow\, x = g y\,\Rightarrow\, y = \color{#c00}{g^{-1}}x\,\Rightarrow\,y\approx x$

$\qquad\ \ \ \ \approx\,$ is $\ \rm\color{#08f}{transitive}\,\ $ by $\,x\approx y\approx z\,\Rightarrow\, y = hz,\, x = g y = g(hz) = (\color{#08f}{gh})z\,\Rightarrow\, x\approx z$

Remark $\,\ G x = \{ gx\ :\ g\in G\}\ $ is called the $G$-orbit of $x.\ $ It is a basic concept in group theory.

A quick way to recognize such group structure is by the subgroup test, i.e. a nonempty $G\subset H$ of a group $H$ forms a group $\iff$ it is closed under division, i.e. $\, g,h\in g\,\Rightarrow g/h = gh^{-1}\in G,\,$ which is clear for $\, G= 7^{\Large \Bbb Z}\ $ since $\, 7^{\large j}/7^{\large k} = 7^{\large j-k}\!\in \Bbb Z\,$ since integers are closed under subtraction. In fact this subgroup test is implicitly used in a complementary form since grade school in inferences like below

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