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I Am studying my teacher's notes on our practice exam, And I am stuck on this part of the question where we are trying to prove that the relation is symmetric.

  1. Symmetry: Let x, y ∈ Q. Suppose xRy. Then ${x = 3^{k} y}$, for some k ∈ Z. Thus, y = $3^{−k}x$, where ${−k ∈ Z}$. Hence, yRx. Therefore, R is symmetric.

I am confused on as to why she made k become -k when she switched x and y. I originally thought that k would stay the same. Can anyone help me understand why she did this?

Jr194
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    You are "confused why she made $k$ become $-k$". Well, she wanted to express $y$ from $x=3^ky$ and hence multiplied the equation by $3^{-k}$. And why? Because we want to write $yRx$ from $xRy$. – Dietrich Burde Nov 13 '19 at 09:18
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    $x \cdot 3^{-k}=(y \cdot 3^k) \cdot 3^{-k}=y \cdot 3^{k-k}=y \cdot 3^0=y \cdot 1=y$. – Mauro ALLEGRANZA Nov 13 '19 at 09:19
  • Thanks for the reply. I had assumed that when proving symmetry, all you have to do is switch the values and see if they are still true. Thus the reason why I got confused when k changed to -k. Now I know that you have to algebraically get there as well. – Jr194 Nov 13 '19 at 09:26

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