Number of one-to-one Function from $\mathbb{N}$ to $\mathbb{N}$

Question

How many one-to-one functions are there from $\mathbb{N}$ to $\mathbb{N}$, Where $\mathbb{N}$ is the set of Natural numbers?

Answer 1

We can count these as follows: any injective function $f:\mathbb{N} \to \mathbb{N}$ is uniquely determined by its image (which is an infinite subset of $\mathbb{N}$), and the order induced on its image via the ordering of the domain (that is, we say $f(1)\le f(2)$, etc). So what we are really asking is how many images are possible, i.e. how many infinite subsets of $\mathbb{N}$ are there (covered here) and how many permutations on the image there are (covered here).

Both of these answers are $\mathbb{R}$, so the answer to your question is the cardinality of $\mathbb{R}\times\mathbb{R}$, which has the same cardinality as $\mathbb{R}$, i.e. the continuum as Sangchul Lee noted in the comments.

Answer 2

Note that $\Bbb N$ has only countably many finite subsets,

Also Note that by Cantor's Theorem $P( \Bbb N)$ has uncountably many elements.

Thus $\Bbb N$ must have uncountably many countably infinite subsets, because otherwise $P( \Bbb N)$ will be countable.

Thus there are uncountably many bijections between $\Bbb N$ and $\Bbb N

Thus there are uncountably many one-one functions between $\Bbb N$ and $\Bbb N$