# Tagged Questions

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### Building a function with codomain equal to a given set of reals.

I was discussing with friends the astounding fact that $\mathbb R$ and the set of real continuous functions were equipotent. I asked for a proof that $\mathbb R$ and $\mathbb R ^{\mathbb R}$ are not ...
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### Covering $\mathbb R^2$ with function graphs

Suppose we have a countable family of function graphs (each function is $\mathbb R\to\mathbb R$, not necessary continuous). Obviously, they cannot cover the whole plane $\mathbb R^2$, because they ...
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### Cardinal of the set of real functions

We know that the cardinal of natural numbers is $\aleph_0$, and the cardinal of real numbers is $\mathfrak c$. Is it correct that the cardinal of real functions is $2^{\mathfrak c}$?
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### Prove: if $A$ is infinite set and $B$ is finite set then $\left| {A - B} \right| = \left| A \right|$

Prove: if $A$ is infinite set and $B$ is finite set then $\left| {A - B} \right| = \left| A \right|$ Well, one way to show it, is to find an injective function, for both directions. First, ...
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### What $P(E)\sim (E \to \{ 0,1\} )$ mean?

i was asked to prove: $P(E)\sim (E \to \{ 0,1\} )$. the left-hand side is the set of all subsets of $E$. (Right?) What about the right-hand side? Thanks.
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### Function composition giving the same value

Let $A = \{f: \mathbb N \rightarrow \mathbb N$ | $\forall n\in \mathbb N \ \ \exists p \ge 1 \ \ f^p(n)=n$} What is $\overline{\overline{A}}$?
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### Does there exist an injection from $P(S)$ to $u(S)$

Let $S$ be an uncountable set , let $u(S)$ denote the set of all uncountable subsets of $S$ and let $P(S)$ denote , as usual , the powerset i.e. the set of all subsets of $S$, then does there exist an ...
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### Prove Cardinality of Power set of $\mathbb{N}$, i.e $P(\mathbb{N})$ and set of infinite sequences of $0$s & $1$s is equal.

I tried doing this by defining a function $f$ which takes an element (a subset of $\mathbb{N}$) and maps it to an infinite sequence of $0$s & $1$s. A subset, i.e the element of $P(\mathbb{N})$ ...
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### Equinumerosity: A Bijection Existence Proof

I'm told that if $m<n$, then the intervals $(0,1)$ and $(m,n)$ are equinumerous. I'm asked to prove this by exhibiting a specific bijection between them. I came up with this: ...
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### Closed Form Cantor Snake Function

Does anyone have the closed form for the Cantor-Snake function and its inverse? By Cantor-Snake, I mean the bijection that maps the Naturals to the Rations - the classic proof that the rationals ...
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### A question regarding the Power set

In the proofs that I have seen so far for showing that the power set $2^X$ of a set $X$ cannot be in bijection to $X$, the common idea is to assume that there exists a surjection $f \colon X \to 2^X$ ...
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### Binary sequences and ${2}^{\mathbb{N}}$ have the same cardinality

I recently got the book "selected problems in real analysis", and I'm stuck solving the very first problem $(u_n)$ is a binary sequence iff it only contains $0$ and $1$ in the sequence Let $A$ be ...
Is there a bijective function $f(n)$, where $n \in \mathbb{N}$, which enumerates all algebraic numbers? Is it possible to define such function?