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I know this problem has something to do with the Cantor-Bernstein Theorem, but how do I show that the set of natural numbers $\mathbb N = \{0,1,2,3,\dotsc\}$ has the same cardinality as the set of non-negative even numbers $E = \{0,2,4,6,\dotsc\}$.

Andrés E. Caicedo
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Ben
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1 Answers1

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To show that two sets have the same cardinality you need to show that there exists a bijection between the sets.

For instance, $f(n)=2n$ is a bijection between $\mathbb N$ and $E=\{0,2,4,\dotsc\}$.

By other hand, Cantor-Bernstein says that we need to find two injective functions: $g\colon\mathbb N\to E$ and $h\colon E\to\mathbb N$. In fact, the functions $f$ and $f^{-1}$ satisfies the conditions.

Cristhian Gz
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