Regarding cardinality and mapping between sets.

In summary, the statement "if ##\vert A\vert\leq\vert B\vert## then there exist an injection from ##A## to ##B##" is not always true, as it is a common mistake in mathematics to use "if" instead of "iff" when defining a concept. This has caused confusion for many in the past.
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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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  • #3
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!
 
  • #4
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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Related to Regarding cardinality and mapping between sets.

What is cardinality and why is it important?

Cardinality refers to the number of elements in a set. It is important in mathematics and computer science because it helps us understand the size and structure of a set, and can be used to compare and classify sets.

What is the difference between one-to-one and onto mappings?

A one-to-one mapping, also known as a bijection, is a function where each element in the domain is paired with exactly one element in the range. An onto mapping, also known as a surjection, is a function where every element in the range has at least one corresponding element in the domain. In other words, a one-to-one mapping is a perfect pairing, while an onto mapping is a complete pairing.

How do you determine if two sets have the same cardinality?

If there exists a one-to-one correspondence between the elements of two sets, then they have the same cardinality. This means that each element in one set is paired with a unique element in the other set, and vice versa.

Can a set have a higher cardinality than its power set?

Yes, it is possible for a set to have a higher cardinality than its power set. For example, the set of real numbers has a higher cardinality than its power set, which is the set of all possible subsets of real numbers.

What is the Cantor-Bernstein-Schroder theorem and how is it related to cardinality?

The Cantor-Bernstein-Schroder theorem states that if there exists a one-to-one mapping between two sets, then those sets have the same cardinality. This theorem helps us compare the sizes of sets and is often used in proofs involving cardinality.

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