What is Numerical Equivalence of Sets and How is it Defined?

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SUMMARY

The numerical equivalence of sets is defined as the existence of a bijective function F: X → Y, indicating that two sets X and Y have the same cardinality. In discussions, it is noted that while some may refer to this concept as "same cardinality," the term "numerical equivalence" is also valid. A single bijective function suffices to establish this equivalence, contrary to the belief that two injective functions are necessary.

PREREQUISITES
  • Understanding of set theory concepts
  • Familiarity with functions, particularly bijective and injective functions
  • Knowledge of cardinality in mathematics
  • Basic grasp of mathematical notation and terminology
NEXT STEPS
  • Research the properties of bijective functions in set theory
  • Explore the concept of cardinality and its implications in mathematics
  • Study examples of numerical equivalence in finite and infinite sets
  • Learn about the differences between injective, surjective, and bijective functions
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Mathematicians, educators, students studying set theory, and anyone interested in the foundational concepts of mathematical equivalence and cardinality.

Piglet1024
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1. Define numerical equivalence of sets
2. I'm not sure how in depth the definition needs to be, how is my current def?
3. X is numerically equivalent to Y if \existsF:X\rightarrowY that is bijective or there are two injective functions f:X\rightarrowY and g:Y\rightarrowX
 
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Seems reasonable. I haven't seen the term "numerical equivalence" used. What I have seen is "same cardinality." I don't think you need to have two injective functions; just one bijective function should do the job.
 

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