Isomorphism as an Equivalence Relation on Sets: A Proof

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SUMMARY

The discussion centers on proving that isomorphism is an equivalence relation on a set of sets, denoted as S. Participants confirm that the proof can be structured using the Reflexive, Symmetrical, and Transitive properties, which are fundamental to equivalence relations. Specifically, they emphasize that sets are isomorphic if they share the same cardinality, thus satisfying the necessary conditions for equivalence. Additionally, there is clarification needed regarding the notation X! and its relation to self-map functions on X.

PREREQUISITES
  • Understanding of equivalence relations in set theory
  • Familiarity with cardinality and its implications for sets
  • Knowledge of mathematical notation, particularly X! and X^x
  • Basic proof techniques in mathematics
NEXT STEPS
  • Study the properties of equivalence relations in detail
  • Explore the concept of cardinality and its role in set theory
  • Research the notation and implications of factorials in set functions
  • Learn about isomorphism in various mathematical contexts, including algebra and topology
USEFUL FOR

Mathematicians, students of abstract algebra, and anyone interested in the foundational concepts of set theory and equivalence relations.

yaganon
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So it says here "Let S be a set of sets. Show that isomorphism is an equivalence relation on S."

So in order to approach this proof, can I just use the Reflexive, Symmetrical, and Transitive properties that is basically the definition of equivalence relations?

eg. suppose x, y, z are sets contained in S...

Reflexive: x~x <same as> xRx
Symmetry: x~y => y~x <same as> xRy => yRx
Transitive: x~y, y~z => x~z

This seems too elementary, but I just want to make sure.
 
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Yes - showing something is an equivalence relation always involves the same steps. Sets are isomorphic if they have the same cardinality, so you need to show that the property of "having the same number of elements" satisfies the three conditions.
 
I'm also confused about the X! notation. I think X^x means the set of all selfmap functions on X. What is X! suppose to be?
 

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