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15

The definition of equicardinal is that there exists a bijection between the sets. You are trying to define "not equicardinal" as "there exists a bijection between one set and a strict subset of another". This definition is not a good one, as all Dedekind infinite sets (such as $\mathbb{Z}, \mathbb{R}$) have the property that they are bijective with strict ...


11

Nice question. I'm not sure that this will be exactly what you want, but let's go for it: consider $X = \{A,B,C,D\}$, and define $d: X \times X \to \Bbb R$ putting $d(p,p) = 0$ for all $p \in X$, $d(p,q) = d(q,p)$ for all $p,q \in X$, and: $$d(A,B) = d(A,C) = d(B,C) = 2, \qquad d(A,D) = d(B,D) = d(C,D) = 1.$$ Here, $1$ and $2$ are for simplicity (note: $1 ...


7

One important set of spaces that are fundamental to analysis are the $L^p$ spaces. For any real number $p \geq 1$, we define $L^p$ to be the set of all (say real-valued) functions $f$ such that $$\int |f(x)|^p dx < \infty$$ where the integral is taken over the domain of interest, for example $\mathbb{R}$. For a function $f \in L^p$, we define the $L^p$ ...


6

The discrete metric is not stupid at all, I do not know why you would say that. (I am not kidding, I bet in the future you will appreciate it more and more :) Take $\{0,1\}$ with, well, not much choice, the discrete metric, $d(0,1)=1$. Now take $\{0,1\}^\omega$ with the product topology. One nice metric for it is $d(\langle x_0,x_1,..\rangle,\langle ...


6

I think completely ruling out subsets of $\,\mathbb{R}^n$ misses the point. Suppose you have a curved surface (for definiteness, sitting in 3-space). If you want to stay inside it, as we do when considering large distances here on earth, we don't use the ambient space metric, but rather the arclength of the shortest path between 2 points, called the ...


6

Here is one that comes up pretty frequently. Consider a graph $(V,E)$ where $V$ is the set of vertices and $E$ the set of edges. For two vertices $v,w\in V$ define $$ d(v,w) = \text{length of shortest path between}~v~\text{and}~w. $$ You can check this is a metric fairly easily. It has applications in discrete mathematics (where you may want to use metric ...


6

Here's a proof of the uniqueness of FTA by induction that is different from the proof you cite from Wikipedia. It is due to Zermelo, and isn't as widely known as the other proofs. Suppose we already proved uniqueness for all numbers $<n$ and are now proving for $n$. If $n$ is prime, there is nothing to prove. Otherwise let $p$ be the smallest divisor of ...


6

It is possible to force this perspective, but it would be incorrect to say that the mathematics "actually" is this way. To be more specific, yes, you can choose to consider the inclusions $j_n\colon \mathbb{R}^n\longrightarrow\mathbb{R}^\infty$ defined by $$j_n(x_1,\ldots,x_n)=(x_1,\ldots,x_n,0,0\ldots)$$ However, the objects $\mathbb{R}^n$ have their own ...


5

One source of free interactive demos is the Wolfram Demonstrations Project. You need to download the freely available Wolfram CDF Player if you don't have Mathematica. For very nice graphs, try the Desmos Graphing Calculator. The output looks really nice. See for instance an interactive example of drawing lines. Try also What's Happening in the Mathematical ...


5

Ok, this is certainly a non-trivially complicated question. I chose to give some sort of geometric intuition. I'm not totally sure this is what you were after, but hopefully it's of some use to you! Geometric prelude So, the best place to start thinking about ramification, is in terms of maps of Riemann surfaces. While this may seem unrelated at first, ...


5

The text A Classical Introduction to Modern Number Theory by Ireland and Rosen discusses unique factorization in Chapter 1. The way they handle unique factorization in $\mathbb{Z}$ is by defining an operator $\operatorname{ord}_pn$ as the number of times a prime $p$ divides $n$ (i.e., $\operatorname{ord}_pn=k$ when $p^kq=n$ and $p\nmid q$). This operator ...


5

As mentioned by others as well, the statement is that $A\sim B \Leftrightarrow \exists \phi:A\leftrightarrow B$ (i.e., $A$ and $B$ are of the same cardinality if there exists a bijection between $A$ and $B$) That is not to say that all maps between them must be bijective, just that there must be at least one such map. Consider $A=\mathbb{N}$ and ...


5

I think the introductory part of Ravi Vakil's FOA book might be helpful. It took me sometime to finish all the execrises carefully, at least. He covered more advanced topics later in the book, which I have not reached so far. It is easy to make a list of topics or reference books(Weibel, Osbourne, Hilton, etc), but I guess the point is to see concrete ...


4

Nothing can be more fertile and instructive than a careful reading through the classic book Categories for the working mathematician by S. Mac Lane. I think the key concept is that of an adjunction and (closely related) of a representable functor, because it literally appears all over the place in algebraic geometry. One motivating example is that the direct ...


4

I think the fundamental error here is a mistaken notion of what the Pigeonhole Principle is. Nowhere in the Wikipedia page cited in the question does it say there is any rule over which hole each pigeon may go into. As is evident from some comments on other answers the argument in the question is based on the misconception that the Pigeonhole Principle ...


4

An example is the Schwartz class $S(\mathbb{R^n}) = \{f(x): \mathbb{R}^n \to \mathbb{C} : f(x) \in C^{\infty}(\mathbb{R}^n)$ and $\operatorname{sup}_{x \in \mathbb{R^n}}|x^{\alpha}\partial^{\beta}f(x)|< \infty$ for all multi-indices $\alpha, \beta$} with the metric $$d\left( f,g \right) =\sum _{\alpha, \beta ...


4

You can make a nice, somewhat abstract argument that connects this; in particular, equilateral triangles have various nice properties - like: For fixed $z_1,z_2$, there are exactly two $z_3$ such that $(z_1,z_2,z_3)$ is an equilateral triangle. If $z_1$ and $z_2$ are real, then those $z_3$ are reflections around the real axis. Any permutation of an ...


4

Your professors are not thinking much differently than you can. But the proofs they are supplying are (most of the time) the results of somebody thinking about the problem "intuitively", seeing why the proposition "has to" be true, then putting each step in that intuition into a justifiable statement. That last step is the one you are having trouble with. ...


4

There is another good source for mathematics outside mathematics world: Feature Column from the AMS. One may try the following categories: History of Mathematics Math and Nature Math and Technology Math and the Arts Math and the Sciences Miscellaneous


3

Here is an extremely good source I have recently found: Panorama And in general, all the page Mathigon is outstanding well done.


3

No, just because a finite dimensional space can be embedded in an infinite dimensional space does not make it "actually infinite dimensional". Dimension is well defined, and preserved under embedding. If you are wondering if any people use standard embeddings of $\Bbb R^n$ into each other (like the conventional $\Bbb N\subset \Bbb Z\subset \Bbb Q\subset\Bbb ...


3

Here's an example of what I mean in my comment. The fact that all sides are equal, and that the angles between them are equal can be expressed quite elegantly as $$\frac{z_3-z_2}{z_2-z_1}=\frac{z_1-z_3}{z_3-z_2}=\frac{z_2-z_1}{z_1-z_3}.$$ The desired equation now follows from some elementary algebra. I believe this result can be generalized to any ...


3

Consider the space of all continuous functions on $[0,1]$ with the sup norm, or the integral norm.


3

The Hilbert space $(z_1,z_2,z_3,\ldots)$ where each $z\in\mathbb C$ and $\sum_{n=1}^\infty |z_n|^2<\infty$ certainly comes up. It is isomorphic to the space of functions $f:[0,1]\to\mathbb C$ with the metric $d(f,g)=\int_0^1 |f(x)-g(x)|^2\,dx<\infty$. Another example is spaces with the uniform metric $d(f,g)=\sup \{ |f(x)-g(x)|: x\in\text{some ...


3

Let's take the path integral around the origin at radius 1, for convenience. On this circle, we have $1/z = \bar{z}$, the complex conjugate of $z$. If $f(z) = 1/z$ then $z \cdot f(z) = 1$ is real, and if $w$ is any complex number on the same ray from the origin as $z$, then $w \cdot f(z)$ is also real. So in particular any infinitesimal step along the ...


2

This answer is specifically about the Pigeonhole Principle, unlike some other answers which have been about infinity. A correct use of the Pigeonhole Principle requires the following: I define what my pigeonholes and pigeons are. The cardinality of the pigeons must be strictly larger than the cardinality of the pigeonholes. My nemesis assigns pigeons to ...


2

Here is an example that comes from number theory rather than from analysis. Euler believed that if you fix a prime number $p$ then infinite series of the form $$a_0 + a_1 p + a_2 p^2 + a_3 p^3 + \cdots $$ make some kind of mathematical sense, where the coefficients are chosen in $\{0,1,…,p-1\}$. More generally one might wonder how to make sense of a similar ...


2

As Bungo said in the comments, you can also consider the $\ell^p(\mathbb{N})$ spaces. For $1 \leq p < \infty$ they're the sets of summable sequences to the p-th power:$$\ell^p = \bigg\{x \colon \sum_{n = 1}^{\infty} |x_n|^p < \infty\bigg\}$$ With the norm $$\|x\|_p = \bigg(\sum_{n = 1}^{\infty} |x_n|^p\bigg)^{1/p} $$ And for $p = \infty$ ...


2

You can get pretty far by visualizing $(d+1)$-dimensional things as movies of $d$-dimensional things moving in time. These kinds of visualizations are closely related to Morse theory. For example, it's a bit tricky to visualize the $3$-dimensional sphere $S^3$, but it's not at all tricky to visualize the following movie: Initially, there is nothing. Then ...


2

By definition, $$e=\lim_{n\to\infty}\left(1+\frac1n\right)^n.$$ Using the binomial theorem, the $k^{th}$ term of the development is $${\binom nk}\frac1{n^k}=\frac{n(n-1)(n-2)\dots(n-k+1)}{k!.n.n.n\dots n},$$ and $$\lim_{n\to\infty}{\binom nk}\frac1{n^k}=\frac1{k!}.$$ For example, ...



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