18
votes
2answers
283 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 ...
2
votes
1answer
37 views

What is the cardinality of Maclaurin series that are entire that induce a bijection between ${\mathbb Z}$ and itself?

So the cardinality of linear polynomials that induce a bijection from ${\mathbb Z}$ to itself is countable, because it is simply the set of linear polynomials of the form $x + n$ where $n$ is an ...
3
votes
3answers
119 views

Cardinality of all sequences of non-negative integers with finite number of non-zero terms. (NBHM 2012)

Consider the set $S$ of all sequences of non-negative integers with finite number of non-zero terms. Is the set $S$ countable or not? What is the cardinality of the set $S$ if it is not countable? ...
1
vote
1answer
62 views

Cardinality of cantor set $K$

Is the Cantor function bijective from $[0,1]$ to Cantor set $K$? As $K$ is uncountable I think cardinality of $K$ must be $\mathfrak c$ as $K$ is a subset of $[0,1]$. But I am surprised whether there ...
2
votes
1answer
146 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 $P(\mathbb{N})$ ...
3
votes
2answers
176 views

What is the cardinality of $[a,b] $?

It is a well-known fact that any open interval $(a,b)$ has the same cardinality as $\mathbb R$; that is, there exists a bijection $f\colon(a,b)\mapsto \mathbb R$. What about the closed interval ...
1
vote
1answer
578 views

minimal infinite sigma algebra [duplicate]

Does there exist sigma algebra whose cardinality is countably infinite? If yes tell me some examples. If not how to show every infinite sigma algebra is uncountable?
2
votes
1answer
52 views

Countability of “center” points of line segments in complement of Cantor set

So, start with the set [0,1] of the real line. Remove the middle third, and keep removing the middle thirds of the remaining line segments as usual when making the Cantor set. Each time you remove a ...
3
votes
7answers
383 views

Proving the uncountability of $[a,b]$ and $(a,b)$

I am trying to prove that $[a,b]$ and $(a,b)$ are uncountable for $a,b\in \mathbb{R}$. I looked up Rudin and I am not too inclined to read the chapter on topology, for his proof involves perfect ...
3
votes
4answers
224 views

Is this set bijective to R?

The set of all infinite sequences with integer entries? Obviously my set is at least as large as R, but is it larger?
3
votes
3answers
240 views

Proving $\mathbb{R}$ is uncountable using Dedekind cuts?

I'm familiar with several proofs that the real numbers are uncountable (Cantor's initial proof, a proof by diagonalization, etc.). However, I've never seen a proof that the reals are uncountable that ...
6
votes
1answer
315 views

On cardinality of equivalence classes

Say $R$ is the set of all Riemann integrable functions $f:[a,b]\rightarrow \mathbb{R}$. We define an equivalence relation in $R$ as follows: $f$ and $g$ are said to be integrally equivalent iff they ...
17
votes
4answers
437 views

What do the cosets of $\mathbb{R} / \mathbb{Q}$ look like?

$\newcommand{\R}{\Bbb R}\newcommand{\Q}{\Bbb Q}$ Looking at the group of real numbers under addition $(\R, +)$ it contains the (normal) subgroup of rational numbers $(\Q, +)$. I am wondering how to ...
7
votes
1answer
423 views

What is the cardinality of a set of all monotonic functions on a segment $[0,1]$?

What is the cardinality of a set of all real monotonic functions on a segment $[0,1]$? Does it really matter that functions are monotonic?
1
vote
3answers
565 views

what is the cardinality of set of all smooth functions in $L^1$?

What is the cardinality of set of all smooth functions belonging to $L^1$ or $L^2$ ? What is that of set of all integrable or square integrable functions ?