I found a question in my textbook but I think the answer provided is wrong.
The question says:
Let $S$ be a relation defined over $\mathbb R$ such that $(a,b) \in S \iff ab≥0$. Is $S$ equivalence?
Book says that it is.
But let $x=4, y=0$ and $z=-6$ be three real numbers. Then clearly $(x,y),(y,z)\in S$ but $(x,z)\notin S$. Therefore $S$ is intransitive, hence not equivalence relation. Am I doing some conceptual blunder? Please help.
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1$\begingroup$ Yes you are right! It is supposed to be strict inequality. $\endgroup$– Madhan KumarCommented Mar 4 at 9:14
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5$\begingroup$ @MadhanKumar A small correction: strict inequality would mean the relation is not reflexive (since $0 \cdot 0 \not > 0$), so it’s still not an equivalence relation. It would need to be restricted to $\mathbb{R}\setminus\{0\}$ (in which case it doesn’t matter if the inequality is strict or not), or the case of $0$ needs to be separately dealt with. $\endgroup$– David GaoCommented Mar 4 at 9:17
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$\begingroup$ yeah right..... $\endgroup$– Madhan KumarCommented Mar 4 at 9:19
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