Intereting Posts

Hint on an exercise of Mathieu groups
How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$?
How do you calculate this sum $\sum_{n=1}^\infty nx^n$?
If $f'(x) = 0$ for all $x \in \mathbb{Q}$, is $f$ constant?
Prove $2^b-1$ does not divide $2^a + 1$ for $a,b>2$
Proving two measures of Borel sigma-algebra are equal
Why is the difference of distinct roots of irreducible $f(x)\in\mathbb{Q}$ never rational?
Identity between $x=y+z$ and $\tan\left(\frac{\theta}{2}\right)=\tan\left(\frac{\nu}{2}\right)\tan\left(\frac{\pi/2-\epsilon}{2}\right) $
Finding a function from the given functional equation .
All real functions are continuous
Bounds for $n$-th prime
Vector Field in a complex projective space
Relationship Between Differential Forms and Vector Fields
How do you solve the cauchy integral equation?
Conjugacy classes of a compact matrix group

Is it correct to say that $\mathbb{R}$ is a subset of $\mathbb{R}^2$? Or, put more generally, given $n,m\in\mathbb{N}$, $n<m$, is $\mathbb{R}^n$ a subset of $\mathbb{R}^m$?

Also, strictly related to that: what is then the “relationship” between the set $\{(x,0)\in\mathbb{R}^2,x\in\mathbb{R}\}\subset\mathbb{R}^2$ and $\mathbb{R}$? Do they coincide (I would say no)? As vector spaces, do they have the same dimension (I would say yes)?

If you could give me a reference book for this kind of stuff, I would really appreciate it.

- what is the relationship between ZFC and first-order logic?
- Question related with partial order - finite set - minimal element
- Relation: pairwise and mutually
- Construct a bijection from $\mathbb{R}$ to $\mathbb{R}\setminus S$, where $S$ is countable
- Proving with diagonal lemma
- Name for collection of sets whose intersection is empty but where sets are not necessarily pairwise disjoint

Thank you very much in advance.

(Please correct the tags if they are not appropriate)

- For all infinite cardinals $\kappa, \ (\kappa \times \kappa, <_{cw}) \cong (\kappa, \in).$
- Distinguishing powers of finite functions
- The empty function and constants
- Can a collection of subsets of $\mathbb{N}$ such that no one set contains another be uncountable?
- Is $\emptyset \in \emptyset$ or $\emptyset \subseteq \emptyset$?
- Sufficient / necessary conditions for $f \circ g$ being injective, surjective or bijective
- Do there exist bijections between the following sets?
- Prove that the interval $(0, 1)$ and the Cartesian product $(0, 1) \times (0, 1)$ have the same cardinality
- Why is the collection of all groups considered a proper class rather than a set?
- How rigorous are pictorial proofs?

It’s not really true that $\mathbb R^n$ is a subset of $\mathbb R^m$ when $n<m$. It is true that there is a subspace of $\mathbb R^m$ that is isomorphic to $\mathbb R^n$, but unfortunately, there are way too many of them, and there is no real way to pick one of them usefully as “the obvious embedding.”

A simple example when $n=1$ and $m=2$ is that the $x-$ and $y-$axes are both embeddings of $\mathbb R^1$ into $\mathbb R^2$, and there is no “natural” way to choose between these enbeddings (or any other embeddings of the real link into $\mathbb R^2$.)

I wouldn’t say so, even though every onedimensional subspace of $\mathbb{R}^n$ is isomorphic to $\mathbb{R}$, but there is no natural embedding.

But a more or less funny is, that even thought nearly everyone say that $\mathbb{R}\not\subset\mathbb{R}^2$ many mathematicans say that $\mathbb{R}\subset\mathbb{C}$

even though there is a canocial transformation from $\mathbb{C}$ to $\mathbb{R}^2$.

I guess sometime one stops to distinguish often between things which are isomorphic but not the same.

Strictly speaking, $\mathbb R$ is not a subset of $\mathbb R^2$.

However, depending on how you construct your number systems, the *natural number* $1$ is different from the *rational number* $1$, and both are distinct from the *real number* $1$. And don’t get me started on complex numbers or quaternions!

This type of imprecision allows us to actually talk about the objects in question without worrying too much about auxiliary details. $\mathbb{R}$ can clearly be embedding into $\mathbb{R}^2$ preserving virtually any property of $\mathbb{R}$ in the process. Once this is realised linguistic license is not too detrimental to rigour.

- Infinite Product Space
- Set of Linear equation has no solution or unique solution or infinite solution?
- How to integrate $\int \frac{dx}{\sqrt{ax^2-b}}$
- If $a$ is even and $b$ is odd then $\gcd(2^{a}+1,2^{b}+1)=1$
- Maximum principle – bounds on solution to heat equation with complicated b.c.'s
- Defining division by zero
- If $X^\ast $ is separable $\Longrightarrow$ $S_{X^\ast}$ is also separable
- What is a local parameter in algebraic geometry?
- A generalization of IMO 1983 problem 6
- Is a probability of 0 or 1 given information up to time t unchanged by information thereafter?
- Texture mapping from a camera image (knowing the camera pose)
- How does $\cos x=\frac12(e^{ix}+e^{-ix})$?
- Stone's Theorem Integral: Basic Integral
- Integration of $x^2 \sin(x)$ by parts
- An arctan integral $\int_0^{\infty } \frac{\arctan(x)}{x \left(x^2+1\right)^5} \, dx$