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7

Probably the most standard term of the sort you're looking for is "equipotent", though it isn't used particularly often (more often, people will just say two sets "have the same cardinality" or "are in bijection"). People who think about categories a lot sometimes say "isomorphic as sets".


7

The Principle of Mathematical Induction says that for all "properties" $P$, $$\left(P(0)\land\forall k\in \mathbb N\left(P(k) \implies P(k+1)\right)\right)\implies \forall n\in \mathbb N(P(n)).$$ So you're basically asking how to write the $\forall k\in \mathbb N\left(P(k)\implies P(k+1)\right)$ bit. It's a universal statement. It's common to start those ...


5

Differentiation under the integral sign gives: $$ \int_{0}^{1}\log(x)\log(1-x)\,dx = \left.\frac{\partial^2}{\partial\alpha\,\partial\beta}\int_{0}^{1}x^\alpha(1-x)^{\beta}\,dx\right|_{\alpha,\beta=0} $$ hence you just have to differentiate a beta function and $\zeta(2)$ arises as $\psi'(1)$. Differentiation is carried on through: $$ \frac{d}{dz}\,f(z) = ...


4

The field of probability can be made mathematically rigorous. Introductory textbooks on probability tend to use terminology like 'sample space', 'outcome', 'event', and outcomes are given names like 'Heads' or 'HTT' or 'King of Spades', all in an attempt to keep things informal and intuitive. These texts often refrain from defining such concepts precisely, ...


3

They knew most of the cutting-edge mathematics of their times, by studying with the people who invented it, or by voraciously reading all they could put their hands on. They maintained extensive contact with colleagues, be it by meeting in some Academy or other or just by mail. Not that far from how today's prominent mathematicians came to know what they ...


3

Well, the simplest solution is just to write up your homework in LaTeX and keep the files in well-organized folders. This also has the advantage of making you quick at LaTeX and giving your professors something far nicer to grade. That said, I have years' worth of old LaTeX documents on old hard drives and I've never seen that it was worth it to go back ...


3

The Blue Eyes Puzzle is a very good candidate in my opinion.


3

First, we should note that a very similar question has already been asked here, and several interesting answers were given. But because the question keeps coming up, I'm going to go out on a limb and suggest that there still might be room for a more complete list of reasons why hyperbolic geometry is important in its own right. It's hard to know where to ...


3

This has to do with the fact that using integration by parts you get a dilogarithm, and $\text{Li}_2(1)=\zeta(2)$, since the two defining series happen to be the same at $x=1$. Why $\zeta(2)$ involves $\pi^2$ can be seen as a by-product of the series expansion of $\sin(x)$, as can be read about here, on the solution to the Basel problem. In short one shows ...


3

Short answer: no. Long answer: noooooooooooooooooooooo. Long serious answer: PDEs are tough. Very tough. And there are two types of PDEs in general: The type we can generalize to ODEs and the type we cannot. When solving PDEs, you consider your work finished when you reduced them to a set of ODEs, because solving the ODEs then should be "trivial". A ...


3

We say "$A$ injects into $B$", and may write $f: A \hookrightarrow B$.


2

I would say yes in general. Of course this depends on what kind of course you take. The huge difference between ODEs and PDEs is that for ODEs one is usualy interested in finding a solution. For PDEs this proves to be way to complicated very fast. There are some simple cases where explicit solutions can be constructed, sometimes by reducing to an ODE. But ...


2

I would have liked to put this as a comment but I do not have enough points to do it. I think there is some magic in this formula because it tells us that the Lie derivative $\mathscr{L}_X$ is homotopic to zero with the homotopy $i_X$ going from top-right $\Omega^{p+1}(M)$ to bottom-left $\Omega^p(M)$ diagonally in the following diagram: $$\require{AMScd} ...


2

This might be a cliché by now but I've always liked the locker problem.


2

I think that many combinatorics problems will be a good candidate, because counting is very intuitive to everyone (and there are very few theorems to be used). A personal favorite is the following: Let $a_n$ be a nondecreasing sequence of integers, and $b_k$ be the number of terms of {$a$} that are greater than $k$. Prove that the sum of the elements of the ...


2

I completely disagree with your statement that it is easier to prove things than it is to find counterexamples. There are statements that are easy to prove For every real valued function $f$, there exists a function $g$ such that $\forall x\in \mathbb R: f(x)<(g)$ and statements hard to prove. There are no integer solutions to the equation ...


2

Module $2$ syllabus can be found in almost all books on Algebra but I recommend Dummit and Foote. For Module $1$, I would recommend "Linear Algebra done right" by Axler for abstract approach avoiding Matrices, and "Linear algebra done wrong " by Sergie Treil (Google it for e-copy). These two books covers all topics espically Sergie's book, but Axler is ...


2

I would go with ACoPS because this book teaches you how to solve hard problems, and gives you ideas and techniques that are quite new that you can use them again or refine them so you can solve a broader class of questions.


1

Before Set Theory (let us take ZFC as our axiomatization in this post) even could be considered as a foundation of mathematics, people had to figure out how to encode many of the objects, mathematics had already dealt with for milenia. Integers, functions, groups, graphs, ... it turned out that sets were a nice way to encode all these things in canonical ...


1

It seems you think your only choices are between pure math and software engineering. I doubt that is true even at an engineering university. If it is true, it seems you have already made your choice. And I have to agree there is no joy in the prospect of spending your one and only life doing something you really don't like. However, I'd like you to ponder ...


1

Becareful that you don't just remember STEPS ON HOW to do it, but WHY THE METHOD WORKS. Once things become common sense and then you practise-practise-practise it's hard to forget. Don't be scared to get it wrong if it helps you understand why it works. Not sure what level maths you are doing but try googling virtualb15 and wootube. NSW syllabus but the ...


1

Take a peek at some no-cost alternatives around the 'net, like Treil's "Linear Algebra done Wrong" (very nice, but might not be all the abstract you'd want). Often lecture notes are available, and add a different explanation that helps you over some rough spot.


1

Reading more mathematics is not going to help you understand more mathematics unless you read with pencil and paper next to you and try to fill the gaps in the proofs that you do not understand or find hard to follow. You don't get better at swimming by watching people swim, you must swim. To get more fluent in mathematics, you must do it as well.


1

I would argue neither is harder in general, or at least there's no objective way of making such a judgement. Finding a counterexample doesn't necessarily require an ingenious guess though, you can find one in a relatively systematic fashion. When proving a theorem, the main question you are trying to answer is "why do these hypotheses imply the conclusion?" ...


1

The deeper we dive into the ocean of mathematics, the stronger is my conviction that we actually do not know anything. Let me give some examples: $1.$ Every polynomial $p(z)$ has a root in $\mathbb{C}$ $2.$ For a Hilbert space $H$, $H=W\oplus W^{\perp}$, where $W$ is a closed subspace of $H$ and $W^{\perp}$ is its orthogonal complement. $3.$ Every subgroup ...


1

The integral gives the area under the graph of a function $f(x)$. However, the integral can also give the average value of a function. Reviewing the definition of the summation definition of the integral, which cuts up the function into rectangles and then sums them, we see that there is a relationship between the summation and the average value formula. ...


1

The $L_p$ norm is more general, but you need to specify a measure space for it to make sense. You're integrating over $\mathbb X$ after all! The $l_p$ norm can be seen as a particular case of the above, as @Stephen Montgomery-Smith noted, with the counting measure on positive integers. So I don't think there really is any source of ambiguity: either I ...


1

This question was asked here some time ago. I answered it yesterday and a kind user told to not link notes, so I take a chance to put some math here. As you can see the formula involves the cardinalities of sets, so it is quite natural to ask if you can bring $G$ to the other side and find an explicit isomorphism. Let's agree on some notation: $X$ is a ...



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