Proving |A|<=Aleph Null with Function f:A->B and |B|<=Aleph Null

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i have a function f:A->B, I am also given that |B|<=null aleph, and for every b in B, |f^-1({b})|<=null aleph, i need to prove that |A|<=null aleph.
basically i think that A equals the union of f^-1({b}) for every b in B, and by another theorem i can consequently assert that |A|<=null aleph.

but is this correct?
 
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Yes...
 
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AKG, thanks.
 

1. How can I prove that |A|<=Aleph Null?

To prove that |A|<=Aleph Null, we can use a function f:A->B and show that |B|<=Aleph Null. This is known as the Cantor-Bernstein-Schroeder theorem.

2. What is the Cantor-Bernstein-Schroeder theorem?

The Cantor-Bernstein-Schroeder theorem states that if there exist injective functions f:A->B and g:B->A, then there exists a bijection between A and B. In other words, if we can find a one-to-one correspondence between the elements of A and B, then |A| and |B| are equal.

3. How does the function f:A->B prove that |A|<=Aleph Null?

The function f:A->B is used to show that there is a one-to-one correspondence between the elements of A and B. This means that for every element in A, there is a unique element in B that it maps to. Since |B|<=Aleph Null, this means that there are at most Aleph Null elements in B, and therefore at most Aleph Null elements in A.

4. Can the Cantor-Bernstein-Schroeder theorem be used to prove other cardinalities?

Yes, the Cantor-Bernstein-Schroeder theorem can be used to prove the equality of any two cardinalities, not just |A| and |B|. It is a powerful tool in set theory and has many applications in proving the equality of infinite sets.

5. Are there other methods to prove |A|<=Aleph Null?

Yes, there are other methods to prove |A|<=Aleph Null, such as using the Axiom of Choice or constructing a bijection directly. However, the Cantor-Bernstein-Schroeder theorem is a commonly used method and can be applied in various scenarios.

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