1
vote
1answer
53 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
78 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
130 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
92 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
60 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 ...
-4
votes
1answer
73 views

Given $f: A \to B$ and $g: B \to A$ prove there exists $h: A \to B$ that is a bijection [duplicate]

Note that $f$ and $g$ need not actually be related. They are both injective though. Please do not use theorems about cardinals (without proof,) since cardinals are actually based on this to begin ...
5
votes
1answer
99 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
200 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
69 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
173 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
71 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
147 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 ...