Undergrad Regarding cardinality and mapping between sets.

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SUMMARY

The discussion centers on the mathematical concept of cardinality and the conditions under which an injection exists between two sets A and B. It is established that the statement "if |A| ≤ |B| then there exists an injection from A to B" is not always true unless explicitly stated as an "if and only if" (iff) condition. Participants noted the common confusion in mathematical definitions where "if" is used instead of "iff," leading to misunderstandings. This clarification is crucial for accurately interpreting cardinality in set theory.

PREREQUISITES
  • Understanding of set theory and cardinality
  • Familiarity with mathematical definitions and notation
  • Knowledge of injections and functions in mathematics
  • Basic comprehension of logical statements and their implications
NEXT STEPS
  • Research the concept of "if and only if" in mathematical definitions
  • Study the properties of injections and their role in set theory
  • Explore examples of cardinality comparisons between finite and infinite sets
  • Learn about the implications of cardinality in advanced mathematics, such as topology
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Mathematicians, students of mathematics, educators, and anyone interested in deepening their understanding of set theory and cardinality concepts.

Terrell
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why is not always true that if ##\vert A\vert\leq\vert B\vert## then there exist an injection from ##A## to ##B##?
 
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Terrell said:
why is not always true that if ##\vert A\vert\leq\vert B\vert## then there exist an injection from ##A## to ##B##?

Who told you that?

By definition ##|A| \leq |B|## iff there exists an injection ##A \to B##
 
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Math_QED said:
Who told you that?

By definition ##|A| \leq |B|## iff there exists an injection ##A \to B##
I think saying that me saying that it's not always true is too strong of a statement, but what really happened is I couldn't find any sources that mentions that it must be an if and only if statement. Thank you a lot! It has cause me a lot of unnecessary thinking lol!
 
Terrell said:
I think saying that me saying that it's not always true is too strong of a statement, but what really happened is I couldn't find any sources that mentions that it must be an if and only if statement. Thank you a lot! It has cause me a lot of unnecessary thinking lol!

That's common in mathematics. When one writes a definition one often says "if" while it should be "iff". I remember it confused me in the beginning too!
 
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Thread 'My basic understanding of set theory'

If there are an infinite number of natural numbers, and an infinite number of fractions in between any two natural numbers, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and... then that must mean that there are not only infinite infinities, but an infinite number of those infinities. and an infinite number of those...
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