0
votes
1answer
16 views

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 ...
1
vote
2answers
47 views

Cardinality of all injective functions from $\mathbb{N}$ to $\mathbb{R}$.

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 ...
4
votes
2answers
121 views

Proving that the cardinality of a set is even

Let $E$ be a set and $f:E\to E$ be a function such that $f\circ f=Id$. Let $A=\{x\in E, f(x)\neq x\}$. Suppose that $A$ is finite. Prove that the cardinality of $A$ is even. My ...
0
votes
1answer
30 views

Find the number of vertices in the graph

Let $n\ge 1$ and $V_n = (\left\{ 1,2,...n \right\}\rightarrow\left\{ 0,1,2 \right\})$. Let us define $G_n = \left<V_n, E_n \right>$. $f,g$, are two vertices. They are connected iff: $$\left|\{ i ...
18
votes
2answers
296 views

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 ...
1
vote
1answer
67 views

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 $2^{\mathfrak c}$?
2
votes
1answer
194 views

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, ...
1
vote
2answers
22 views

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.
1
vote
3answers
46 views

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 $\overline{\overline{A}}$?
1
vote
1answer
43 views

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 ...
2
votes
1answer
177 views

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 $P(\mathbb{N})$ ...
0
votes
1answer
105 views

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: ...
0
votes
1answer
77 views

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 ...
5
votes
1answer
104 views

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 $f \colon X \to 2^X$ ...
1
vote
1answer
268 views

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 ...
1
vote
1answer
71 views

Bijection for algebraic numbers

Is there a bijective function $f(n)$, where $n \in \mathbb{N}$, which enumerates all algebraic numbers? Is it possible to define such function?
2
votes
1answer
175 views

Bijection from $(0,1]$ to $[0, \infty)^2$

Define a bijection from $(0,1]$ to $[0, \infty)^2$ Route to follow, A-) First define a bijection from $(0,1]$ to $(0,1]^2$ B-) Since there is a bijection from $(0,1]$ to $[0, \infty)$, namely $f(x) ...
4
votes
1answer
73 views

$|S_X|=|S_Y| \Leftrightarrow |X|=|Y|$

Reading this problem I remembered trying to solve the following problem. For a set $A$, denote by $S_A=\{ f : A \to A | f \text{ is bijective }\}$. Denote by $|X|$ the cardinal number of $|X|$. ...
4
votes
2answers
156 views

Number of continuous $[0; 1] \to [0; 1]$ functions for given arc length

Just out of pure curiosity ... Suppose I want to connect the two points $(0,0)$ and $(1,1)$ with the graph of some continuous and differentiable function $$f : [0; 1] \to [0; 1]$$ and let $s$ be ...