Do Infinite Sets K Have Equal Cardinalities for Permutations and All Functions?

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SUMMARY

The discussion centers on the cardinality of the set of permutations of an infinite set K compared to the set of all functions from K to itself. It is established that the set of permutations on an infinite set K has the same cardinality as all functions from K to K. The participants emphasize the importance of establishing a bijection to prove this equivalence. Additionally, they reference the specific case of K = {a, b} to illustrate the concept of permutations and functions.

PREREQUISITES
  • Understanding of cardinality in set theory
  • Familiarity with permutations as one-to-one and onto functions
  • Knowledge of bijections and their role in proving set equivalences
  • Basic concepts of infinite sets in mathematics
NEXT STEPS
  • Research the concept of bijections in set theory
  • Explore the properties of infinite sets and their cardinalities
  • Study the differences between permutations and general functions
  • Investigate examples of cardinality comparisons for finite and infinite sets
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Mathematicians, students of set theory, and anyone interested in the properties of infinite sets and their cardinalities.

moo5003
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Is it true that the set of permutations on a infinite set K has the same cardinality as all functions between a infinite set K to itself?
 
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Did you try to set up a bijection? Or did you try anything to (dis)prove what you're aksing for?
 
To add to what Pere Callahan said: How many permutations are there on K= {a, b}?
How many functions are there from K to K?
(Permutations are one-to-one and onto functions.)
 

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