Check if relation is equivalent

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    Equivalent Relation
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SUMMARY

The relation defined as \(m \sim n\) in \(\mathbb{Z}\) if \(mn > 0\) is confirmed to be not an equivalence relation. The primary reason is the failure of reflexivity, as the pair \((0,0)\) does not satisfy the condition \(mn > 0\). Therefore, this relation does not meet the criteria for equivalence relations, which require reflexivity, symmetry, and transitivity.

PREREQUISITES
  • Understanding of equivalence relations in mathematics
  • Familiarity with reflexivity, symmetry, and transitivity properties
  • Basic knowledge of integer properties in \(\mathbb{Z}\)
  • Ability to interpret mathematical notation and relations
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  • Study the properties of equivalence relations in detail
  • Explore examples of equivalence relations in different mathematical contexts
  • Learn about non-equivalence relations and their implications
  • Investigate the role of reflexivity in defining relations
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Mathematicians, students studying abstract algebra, and anyone interested in the properties of relations in set theory.

issacnewton
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Hello

I have to check if the following relation is an equivalence relation.
\[m\sim n \;\;\mbox{in}\;\;\mathbb{Z}\;\;\mbox{if}\; mn > 0\]

I think this relation fails to be reflexive since $(0,0)$ does not belong to
this relation. Hence this is not an equivalent relation. Is this ok ?

Thanks
 
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IssacNewton said:
Hello

I have to check if the following relation is an equivalence relation.
\[m\sim n \;\;\mbox{in}\;\;\mathbb{Z}\;\;\mbox{if}\; mn > 0\]

I think this relation fails to be reflexive since $(0,0)$ does not belong to
this relation. Hence this is not an equivalent relation. Is this ok ?

Thanks
Yes. This is good.
 
caffeinemachine said:
Yes. This is good.

Thanks
(Emo)
 

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