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I am trying to understand how to determine the equivalence classes. I have this question in the book

The relation $S$ on the set $\mathbb{R}^*$ is definied as $aSb \iff ab > 0$

The answer in the book is $[1] = \mathbb{R}^+$ and $[-1] = \mathbb{R}^-$.

I don't even know where to start with this problem. I know that $ab > 0$ so then $a \cdot b$ always have to be positive and I know that equivalence class means disjoint categories but why are $[1]$ and $[-1]$ the equivalence classes?

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    $\begingroup$ "I know that ab>0 so then a⋅b always have to be positive". $\endgroup$ – user247327 Mar 29 at 12:59
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First, for all $a,b \in \mathbb{R}^-$, $ab > 0$. So all the elements in $\mathbb{R}^-$ are in the same equivalence class.

Secondly, for all $a,b \in \mathbb{R}^+$, $ab > 0$. So all the elements in $\mathbb{R}^+$ are in the same equivalence class.

Finally, if $a \in \mathbb{R}^-$ and $b \in \mathbb{R}^+$, $ab < 0$. So $a$ and $b$ are not in the same equivalence class.

This shows you that the two equivalence classes (containing $\mathbb{R}^-$ and $\mathbb{R}^+$) are disjoint, so you have indeed two equivalent classes : $\mathbb{R}^-$ and $\mathbb{R}^+$.

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  • $\begingroup$ I thought $[1]$ and $[-1]$ where the equivalent classes? $\endgroup$ – Sam Mar 29 at 13:02
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    $\begingroup$ Yes, but $[1]$ denotes the equivalence class in which $1$ belongs (i.e. $\mathbb{R}^+$), and $[-1]$ denotes the equivalence class in which $-1$ belongs (i.e. $\mathbb{R}^-$). $\endgroup$ – TheSilverDoe Mar 29 at 13:04
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Numbers $a$ and $b$ are equivalent with respect to relation $R$ if their product is positive. If you take $a>0$ what numbers are in relation with $a$? All $b>0$. Similarly for $a<0$, $b$ is in relation with $a$ if $ab>0$, so $b$ must be negative. $R$ partitions $\mathbb{R}^*$ into two nonempty, disjoint sets which add up to the whole $\mathbb{R}^*$, so the two equivalence classes are $\mathbb{R}_+$ and $\mathbb{R}_-$. Any positive number is in relation with $1$, so we can write $[1]_R=\mathbb{R}_+$, and similarly, we can denote $\mathbb{R}_-=[-1]_R$.

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