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What is the cardianlity of: $$ A = \left\{ f:\mathbb{N}\to\mathbb{R} : \text{f is injective} \right\} $$

Trying to prove it using Cantor–Bernstein–Schroeder theorem, I have the obvious side: $$A \subseteq f:\mathbb{N}\to\mathbb{R}$$

Hence, $$\left|A\right| \le \aleph$$

I need to find an injection from a set with cardinality of $\aleph$ to $A$, but couldn't think of a proper one. It's tricky.

Any idea?

Thanks.

Elimination
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2 Answers2

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For each $a\in(0,1)$ we'll define a function $f(n) = a+n$. $f$ is clearly in $\mathbb{N}\to\mathbb{R}$ and since it's monotone it is also injective. Moreover, if $a_1\ne a_2$ then $f_1(n) \ne f_2(n)$.

Hence, the set of all functions described is subset of $A$ and it's cardinality is $\left|\left(0,1\right)\right| = \aleph$

Elimination
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  • Yeah, that's also works. (Simpler than my hint; although proving that the set of injections from $\Bbb N$ to itself has size $\aleph$ gives you more information.) – Asaf Karagila Sep 08 '14 at 17:30
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HINT: Prove that $\{f\in A\mid\operatorname{range}(f)\subseteq\Bbb N\}$ has size $\aleph$.

Asaf Karagila
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    Or just ask it the other way round (injections N -> R). – Martin Brandenburg Sep 08 '14 at 17:19
  • Right. I'll claim it was a typo! – Elimination Sep 08 '14 at 17:19
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    @Elimination: Alright, a different hint, then. – Asaf Karagila Sep 08 '14 at 17:26
  • Can I get an hint for the hint? :) – Elimination Sep 08 '14 at 17:37
  • @Elimination: Certainly. For every infinite set $D\subseteq\Bbb N$ consider the function which fixes pointwise $\Bbb N\setminus D$, partitions $D$ into pairs and switches all those pairs. – Asaf Karagila Sep 08 '14 at 17:40
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    I dont get it :| – Elimination Sep 08 '14 at 18:43
  • @Elimination: Pick any bijection $h\colon\Bbb{N\to N}$ without fixed points (for example, exchange the $n$-th even number with the $n$-th odd number). Now given an infinite subset of $A\subseteq\Bbb N$, consider the function $f_A\colon\Bbb{N\to N}$ which does this: $f_A(n)=n$ whenever $n\notin A$; if $n=a_k$, then $f_A(n)=a_{h(k)}$. (Where $a_k$ is a fixed enumeration of $A$, e.g. the one which respects the natural order of the natural numbers.) Now show that $f_A$ is injective, and that $A\neq B$ implies $f_A\neq f_B$. And how many infinite subsets does $\Bbb N$ have? – Asaf Karagila Sep 08 '14 at 18:45