Equivalence Relation on ℝ: xRy if x≥y | Symmetry and Transitivity Explained

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SUMMARY

The relation defined as xRy if x≥y on the set of real numbers ℝ is not an equivalence relation. While it satisfies the reflexive property (x≥x for all x), it fails to meet the criteria for symmetry and transitivity. Specifically, symmetry is violated because if x≥y, it does not imply that y≥x unless x equals y. Transitivity is also not satisfied, as there exist values where x≥y and y≥z do not lead to x≥z. Therefore, the partition arising from this relation does not form equivalence classes.

PREREQUISITES
  • Understanding of equivalence relations in mathematics
  • Familiarity with reflexive, symmetric, and transitive properties
  • Basic knowledge of real numbers (ℝ)
  • Ability to analyze mathematical statements and proofs
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  • Study the properties of equivalence relations in detail
  • Explore examples of equivalence relations on ℝ, such as xRy if x=y
  • Learn about partitions of sets and their relation to equivalence relations
  • Investigate counterexamples to equivalence relations for better understanding
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Homework Statement



Determine whether the given relation is an equivalence relation on the set. Describe the partition arising from each equivalence relation:

xRy in ℝ if x≥y

Homework Equations



Reflexive: for all x in X, x~x
Symmetric: for all x,y in X, if x~y, then y~x
Transitive: for all x,y,z in X, if x~y, and y~z, then x~z

The Attempt at a Solution



I showed that it's reflexive, because x≥x

I'm kind of confused in regards to how to show that it's symmetric and transitive (if it is)?

Any help is appreciated, thanks.
 
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The first thing i would do it's to try some value of x y and z to see if the property holds.
 

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