Intereting Posts

A “number” with an infinite number of digits is a natural number?
What is the relationship between base and number of digits?
Finding the ring of integers of $\mathbb Q$ with $\alpha^5=2\alpha+2$.
Example of non-trivial number field
Examples of 2-dimensional foliations of a 4-sphere.
Does $\lfloor \sqrt{p} \rfloor$ generate all natural numbers?
Infinite linear independent family in a finitely generated $A$-module
Subtracting a constant from log-concave function preserves log-concavity, if the difference is positive
Why does the Fibonacci Series start with 0, 1?
Question about Infimum. (conclusion about intersection)
Derivative of an even function is odd and vice versa
Inequality involving expectation
Predict next number from a series
$\mathfrak{a}(M/N) = (\alpha M + N) / N$
Partial Proof of Second Hardy-Littlewood Conjecture (modified)?

Prove that the interval $(0, 1)$ and the Cartesian product $(0, 1) \times (0, 1)$ have the same cardinality

using the SB theorem?

Also how does one find a bijection on $f:(0, \infty) \to (0,1)$ such that they have the same cardiniality?

- $\bigcup \emptyset$ is defined but $\bigcap \emptyset$ is not. Why?
- The number of subsets of a set of cardinality $n$
- Countable set having uncountably many infinite subsets
- Proof that $\bigcap_{a\in A} \bigg(\, \bigcup_{b\in B} F_{a,b} \, \bigg) = \bigcup_{f\in ^AB} \bigg(\, \bigcap_{a \in A} F_{a,f(a)}\,\bigg) $
- Easiest way to prove that $2^{\aleph_0} = c$
- What is a null set?
- Taking away infinitely many elements infinitely many times
- What means a “$\setminus$” logic symbol?
- Is every subset of $\mathbb{Z^+}$ countable?
- substituting a variable in a formula (in logic)

Answer to **second question** (much easier):

Take for example $f(x)=e^{-x}$

Answer to **first question**:

There are some obvious injective functions from $(0,1)$ to $(0,1)\times(0,1)$, namely $f(x)=(\frac12,x)$, for example.

So the difficult part is finding an injective function from $(0,1)\times(0,1)$ to $(0,1)$. But it is not *so* difficult. In decimal expansions, let’s disallow infinite strings of $9$’s and we have an unique decimal expansion for each number in $(0,1)$. If we have two numbers like $0.a_1a_2a_3\ldots$ and $0.b_1b_2b_3\ldots$, we can “build” a number merging alternatively the digits of each one, so we define

$$f(0.a_1a_2a_3\ldots, 0.b_1b_2b_3\ldots)=0.a_1b_1a_2b_2a_3b_3\ldots$$

An injection $(0,1)\to(0,1)\times(0,1)$ is obvious; you nead an injection $(0,1)\times(0,1)\to(0,1).$

First, we have injections

$$(0,1)\to\mathcal P(\mathbb Q)\to\mathcal P(\mathbb N).\tag1$$

For the injection $(0,1)\to\mathcal P(\mathbb Q)$ use $x\mapsto\{q\in\mathbb Q:q\lt x\}.$

For the injection $\mathcal P(\mathbb Q)\to\mathcal P(\mathbb N),$ use the fact that $|\mathbb Q|=|\mathbb N|.$

Let $D=\{n\in\mathbb N:n\text{ is odd}\}$ and $E=\{n\in\mathbb N:n\text{ is even}\}.$ Then we have injections

$$(0,1)\times(0,1)\to\mathcal P(\mathbb N)\times\mathcal P(\mathbb N)\to\mathcal P(D)\times\mathcal P(E)\to\mathcal P(\mathbb N)\to(0,1).$$

For the injection $(0,1)\times(0,1)\to\mathcal P(\mathbb N)\times\mathcal P(\mathbb N)$ use (1).

For the injection $\mathcal P(\mathbb N)\times\mathcal P(\mathbb N)\to\mathcal P(D)\times\mathcal P(E)$ use the fact that $|D|=|E|=|\mathbb N|.$

For the injection $\mathcal P(D)\times\mathcal P(E)\to\mathcal P(\mathbb N)$ use $(X,Y)\mapsto X\cup Y).$

For the injection $\mathcal P(\mathbb N)\to(0,1)$ use

$$X\mapsto\frac14+\sum_{x\in X}\frac1{3^{x+1}}.$$

There is a bijection between $(0,1) $ and $ \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) $ by the following map $$ x\mapsto \pi x-\frac{\pi}{2} $$ and $\tan(x)$ is a bijection between $\mathbb{R}$ and $ \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) $. So we take the composition to get the bijection between $(0,1)$ and $\mathbb{R}.$ In the first answer you will get the bijection from $(0,1)\times (0,1) $ to $(0,1)$. The following link proved that cardinality of $\mathbb{R}$ and $\mathbb{R}^2$ are same. $(0,1)\times (0,1)$ ahs the same cardinaltiy as $\mathbb{R}^2.$ So $(0,1)$ and $(0,1)\times (0,1)$ have the same cardinality.

For the second part, $$ f(x)=\frac{x}{1+|x|},\ x\in (0,\infty) $$

- Some basic book to start with modules?
- To show that either $R$ is a field or $R$ is a finite ring with prime number of elements and $ab = 0$ for all $a,b \in R$.
- Evaluating $\lim _{x\to 1}\left(\frac{\sqrt{x}-1}{2\sqrt{x}-2}\right)$
- Why is a projection matrix symmetric?
- Show that a connected graph on $n$ vertices is a tree if and only if it has $n-1$ edges.
- Subset of the preimage of a semicontinuous real function is Borel
- Is there a measurable set $A$ such that $m(A \cap B) = \frac12 m(B)$ for every open set $B$?
- Computing the volume of a region on the unit $n$-sphere
- Evaluating $\int_a^b \frac12 r^2\ \mathrm d\theta$ to find the area of an ellipse
- Understanding the ideal $IJ$ in $R$
- What is the name for defining a new function by taking each k'th term of a power series?
- How to prove that $f (x) = \mu (x + B)$ is measurable?
- Is there a bijection between $\mathbb N$ and $\mathbb N^2$?
- Smooth Functions with Compact Support and Bounded Derivatives of Low Order
- Compute the definite integral