Infinite Sets Recap

Definition. A set $C$ is countable if and only if there exists a surjective function from $\mathbb{N}$ to $C$. (That is, $\le 1$ arrow out from $\mathbb{N}$, $ge 1$ arrow in to $C$.)

Definition. A set $C$ is countably infinite if and only if there exists a bijection between $C$ and $\mathbb{N}$.

Cantor’s Theorem

For all sets, $S$, $| pow(S) | > | S |$.

What does this mean for $| \mathbb{N} |$?

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What is a real number?

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Show there is a bijection between $[0, 1)$ and $pow(\mathbb{N})$.

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