Equivalence between a set and the subset of its subset

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SUMMARY

The discussion centers on the equivalence between a set A and its subset A1, specifically addressing the conditions under which A is not equivalent to any subset of A1. It is established that if the equivalence relation used is based on the existence of a bijection, then the statement holds true. However, alternative equivalence relations, such as having the same parity, can lead to a false conclusion. The conversation highlights the importance of the chosen equivalence relation in determining the validity of the statement.

PREREQUISITES
  • Understanding of equivalence relations
  • Knowledge of bijections in set theory
  • Familiarity with Cantor-Bernstein theorem
  • Concept of parity in mathematics
NEXT STEPS
  • Study the properties of equivalence relations in depth
  • Explore the concept of bijections and their implications in set theory
  • Review the Cantor-Bernstein theorem and its applications
  • Investigate different types of equivalence relations, including parity
USEFUL FOR

Mathematicians, educators, and students studying set theory and equivalence relations, particularly those interested in the implications of different equivalence relations on set equivalence.

batballbat
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Is it true that: If A is not equivalent to its subset A1. Then A is not equivalent to any subset of A1?
 
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You should consider also informing us about the relevant equivalence relation, as the truth or falsity of your statement heavily depends on that information.

For example, if you for your eq.rel. use existence of a bijection, the statement is true, but other relations, like having same parity, will render your statement false.
 
actually this is trivial. I just learned all this the complicated way, .i.e. proving cantor bernstein without the well ordering theorem.
 

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