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I'm looking for an injective function from the set $A$ of all functions $f: \mathbb{R} \to \mathbb{R}$ to $\mathcal{P}(\mathbb{R})$. Any hints?

I think the other direction is easy: An injective function from $\mathcal{P}(\mathbb{R})$ to $A$ is just a functions that maps all $X \in \mathcal{P}(\mathbb{R})$ to a function $g$ with $g(\mathbb{R})=X$. Is that correct?

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How are you going to pick that function $g$? You need to specify a definite rule (and you’ll have a hard time finding a $g:\Bbb R\to\Bbb R$ such that $g[\Bbb R]=\varnothing$, though your idea can be modified to avoid that problem). An easier approach: send $X$ to the indicator function of $X$. –  Brian M. Scott Nov 10 '12 at 18:05
    
I am fairly certain that I posted an answer to this question at least twice on this site. One of them quite recently. Please try to search before asking. –  Asaf Karagila Nov 10 '12 at 18:32
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2 Answers

up vote 2 down vote accepted

How are you going to pick that function $g$? You need to specify a definite rule (and you’ll have a hard time finding a $g:\Bbb R\to\Bbb R$ such that $g[\Bbb R]=\varnothing$, though your idea can be modified to avoid that problem). An easier approach: send $X$ to the indicator function of $X$.

For an injection in the other direction, note that a function from $\Bbb R$ to $\Bbb R$ is a subset of $\Bbb R^2$: it’s a set of ordered pairs of real numbers. Thus, all you really need is an injection from $\wp(\Bbb R^2)$ to $\wp(\Bbb R)$. From an earlier question you already have an injection from $\Bbb R^2$ to $\Bbb R$; use it to get an injection from $\wp(\Bbb R^2)$ to $\wp(\Bbb R)$

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Hint. Note that $\mathbb R \cong \mathcal P(\mathbb N)$, and that every function $\mathbb R \to \mathcal P(\mathbb N)$ corresponds uniquely (in a natural way) to a subset of $\mathbb R \times \mathbb N$.

Now you only need $\mathcal P(\mathbb R\times \mathbb N)\cong \mathcal P(\mathbb R)$ ...

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