1
vote
2answers
53 views

Proof - Inverse of linear function is linear

This is my first proof related to linear functions. It refers to the linear-algebra-$\textit{linear}$ (not the calculus-$\textit{linear}$). Please comment. Theorem The inverse of a linear bijection ...
0
votes
2answers
82 views

Fields - Proof that every multiple of zero equals zero

This is one of my first proofs about fields. Please feed back and criticise in every way (including style and details). Let $(F, +, \cdot)$ be a field. Non-trivially, $\textit{associativity}$ implies ...
15
votes
16answers
1k views

Beautiful, simple proofs worthy of writing on this beautiful glass door [closed]

What are some of the more beautiful proofs you know? I am measuring beauty in two dimensions -- first, how conceptually elegant is it and second, how aesthetically pleasing is it. Context: I work ...
3
votes
1answer
144 views

Proof that group is commutative if every element is its inverse (feedback wanted)

This is one of my first proofs about groups. Please feed back and criticise in every way (including style & language). Axiom names (see Wikipedia) are italicised. $e$ denotes the identity element. ...
3
votes
1answer
49 views

Groups - Prove that every element equals inverse of inverse of element

This is my first proof about groups. Please feed back and criticise in every way (including style & language). Axiom names (see Wikipedia) are italicised. We use $^{-1}$ to denote inverse ...
1
vote
1answer
44 views

Do you paragraph a proof?

When writing out a proof of moderate length, i.e. a proof taking less than or equal to 5 A4 papers and with normal spacing (please avoid asking the criterion for "normal"), do you tend to paragraph it ...
7
votes
3answers
95 views

How to structure long proofs

How do you structure proofs that are longer than say half a page? I have already encountered a variety of styles (in my short math life), some of which I list below and I just hoped to hear some wise ...
7
votes
4answers
665 views

The role of 'arbitrary' in proofs

Generally, when one is going to prove a result regarding a set of elements, they begin their proof with those first few pleasing words: "Suppose...is an arbitrary element in..." My question is, why ...
38
votes
7answers
2k views

When working proof exercises from a textbook with no solutions manual, how do you know when your proof is sound/acceptable?

When working proof exercises from a textbook with no solutions manual, how do you know when your proof is sound/acceptable? Often times I "feel" as if I can write a proof to an exercise but most ...
1
vote
3answers
117 views

Are truth tables a valid method to prove an iff statement?

I recently had a homework assignment returned to me (for a Differential Geometry course, undergrad level) in which my instructor wrote "You cannot use truth tables to prove an if and only if ...
3
votes
1answer
90 views

Best proof of some theorems in calculus

I would like to choose (among the miriads of proofs) a well-structured, elegant, neat, clear proof of the first fundamental theorem of calculus; the second fundamental theorem of calculus; the mean ...
1
vote
1answer
65 views

My first proof employing the pigeonhole principle / dirichlet's box principle - very simple theorem on real numbers. Please mark/grade.

What do you think about my first proof employing the pigeonhole principle? What mark/grade would you give me? Besides, I am curious about whether you like the style. Theorem Among three elements ...
0
votes
2answers
39 views

Should I create two distinct proofs? [*Soft question*]

This is a soft question, and if it is of poor quality, just let me know. As a method of improving my proofing abilities, should I make it habit to go about proving something twice. What I mean by ...
1
vote
1answer
45 views

How to show whether a statement is true or false(Example question inside)?

So I'm reading How to Read and Do Proofs by Solow and I'm on the exercises now. So far it has been good but I'm stuck on how to answer a question. There are no answers for even numbered questions in ...
1
vote
1answer
69 views

Possible book correction or am I missing something?

Hi I am teaching myself analysis and bought "Analysis - With an introduction to Proof" by Steven R. Lay. Now one of the practice problems is "Determine the truth value of each statement, assuming x, y ...
13
votes
7answers
1k views

How do we know whether certain mathematical theorems are circular?

There are countless mathematical theorems and lemmata, some of which, obviously, depend on others. My question is: how do we know that, say, Theorem $A_1$- which uses a result proved in Theorem $A_2$ ...
3
votes
2answers
120 views

When do we write “we are done”?

This may seem like a bit of a silly question, but I notice that in some proofs (a remarkable amount), the author writes: "We are done." after completing a proof. Is this the equivalent of writing one ...
15
votes
3answers
663 views

Starting sentences with mathematical symbols.

I apologise if this is a duplicate in any way or is too opinion-based. To what extent is it best not to start a sentence with a mathematical symbol? I find that when trying to solve a problem or ...
20
votes
7answers
2k views

LaTeX/TeX Vs. Mathematica for Typesetting

I know Mathematica like the back of my hand, but I do not know a speck of $\LaTeX$ or $\TeX$. With regards to mathematical typesetting, is there something significant I can do in $\LaTeX$/$\TeX$ that ...
3
votes
2answers
190 views

Beautiful proof for $e^{i \pi} = -1$ [closed]

To celebrate the recent neuroscientific study that shows the beauty of math is in the mind, what is your most beautiful proof that $e^{i \pi} = -1$?
4
votes
2answers
369 views

Definition: Theorem, Lemma, Proposition, Conjecture and Principle etc.

Definition: Theorem, Lemma, Proposition, Corollary, Postulate, Statement, Fact, Observation, Expression, Fact, Property, Conjecture and Principle Most of the time a mathematical statement is ...
67
votes
14answers
7k views

Why are mathematical proofs that rely on computers controversial?

There are many theorems in mathematics that have been proved with the assistance of computers, take the famous four color theorem for example. Such proofs are often controversial among some ...
3
votes
1answer
114 views

Induction Proofs in Abstract Algebra

In several abstract algebra textbooks, I have been seeing propositions that I would think require induction verified without using induction. For example, consider the claim that if $G_{1}, \ldots, ...
3
votes
0answers
68 views

Proofs involving Disjunctions [Velleman, Chapter 3.5]

$\Large{{1.}}$ Are proofs using strategies $P136, P143$ always easier than those using $P140$? In the former two, only one statement (either $P$ or $Q$) must be proven. In the latter, both $P$ and ...
13
votes
4answers
364 views

Tips for writing proofs

When writing formal proofs in abstract and linear algebra, what kind of jargon is useful for conveying solutions effectively to others? In general, how should one go about structuring a formal proof ...
8
votes
3answers
134 views

What to look for in a proof?

I am a physics undergrad, wishing to pursue a PhD in Math. I am mostly self taught in the typical math undergrad curriculum. I am looking for more input, in ways I can improve my mathematical ...
10
votes
6answers
312 views

How to explain that proof is important

I don't know if this is the right place to post this or not, but I will go ahead anyway (sorry if it ain't the right place) Yesterday I was discussing a particular theorem of geometry with my brother ...
3
votes
3answers
143 views

Still struggling with proofs. [closed]

How do you construct rigorous math proofs on your own? Also how do you verify? I am finishing up my first semester of undergraduate analysis and still am struggling with writing proofs. Even though I ...
0
votes
1answer
115 views

Show a subring contains certain elements.

Show that the set of all real numbers of the form $a_0 + a_1\pi + a_2\pi^2 +\cdots+ a_n\pi^n$ with $n≥0$ and $a_i ∈ \mathbb{Z}$ is a subring of $R$ that contains $\mathbb{Z}$ and $\pi$. ...
5
votes
6answers
378 views

“$n$ is even iff $n^2$ is even” and other simple statements to teach proof-writing

I am supposed to teach undergraduate students who do not major in mathematics and I would like to give them a short introduction to mathematical reasoning and to the concept of proof. I am looking for ...
0
votes
2answers
39 views

How to introduce cases in a proof.

I am writing a simple proof in regards to a homework problem about a property of stable matching. The content of the proof isn't necessarily what I am asking about as opposed to the presentation of ...
7
votes
1answer
302 views

How to write well in analysis (calculus)?

This is kind of a subjective question, I know; often I find myself failing exams and homeworks because of the way i write down proofs. Either I don't know how to start, or somehow the main point of ...
25
votes
5answers
936 views

Level of Rigor in Mathematical Physics

I am a physics/math undergrad and I have recently become familiar with some more rigorous formalisms of mechanics, such as Lagrangian mechanics and Noether's Theorem. However, I've noticed that the ...
4
votes
4answers
411 views

Good book for learning and practising axiomatic logic

I want to learn axiomatic (Hilbert style ) logic. not just a book that says that it exist and is an good way to proof theorems. What is a good book to learn and practice this method? would like: - a ...
4
votes
0answers
193 views

Good examples of proofs in mathematics exemplary of creative reasoning [closed]

Just what the title says. I'm not looking for any proofs that require specialized knowledge past the very fundamentals of real analysis. I'm looking for proofs for important results (don't have to be ...
26
votes
2answers
696 views

English words in written mathematics

I recently marked over $100$ assignments for a multivariable calculus course. One question which a lot of people did poorly was proving a given set was open. Aside from issues relating to rigour and ...
3
votes
1answer
145 views

Providing a sketch for a proof before proceeding through the actual proof. [closed]

Question is pretty straightforward. My mathematics is sloppy, and I recognize my inaptitude in that my proofs are more or less too intuitive. My diagnosis dictates the fact that I attack a problem ...
5
votes
5answers
1k views

How does one begin to even write a proof?

I'm in my first proof based class and I'm just having a lot of trouble writing proofs. I mean I know it's not going to come natural and it will take time, but seroiusly, how does someone begin to ...
37
votes
7answers
3k views

How does a non-mathematician go about publishing a proof in a way that ensures it to be up to the mathematical community's standards?

I'm a computer science student who is a maths hobbyist. I'm convinced that I've proven a major conjecture. The problem lies in that I've never published anything before and am not a mathematician by ...
5
votes
3answers
223 views

In a proof that is reliant on proven theorems, does one assume the reader's familiarity with said theorems, or explicitly include their logic?

In composing a proof that is reliant on proven theorems, does one simply assume the reader's familiarity with said theorems, or does one explicitly include their logic in the new logic?
8
votes
4answers
414 views

Should one imagine diagrams/figures when working?

I'm working through Baby Rudin and find it exceedingly difficult to understand what's happening without drawing a small figure. For instance when proving properties of compactness, I would often draw ...
5
votes
6answers
616 views

How to make sure a proof is correct

If you come up with a proof of a mathematical proposition, how do you verify the proof is correct? Put it another way, how do you avoid a wrong proof? I guess there is no definitive answer to this. ...
15
votes
4answers
671 views

Advice for writing good mathematics?

It's been a (far-fetched, possibly) goal of mine to some day write a math Textbook. I've been thinking about writing this question for a while, but reading an exceedingly mediocre text on Mathematical ...
1
vote
0answers
101 views

What is the convention for using results of theorems left as exercise in the text?

I'm working on an exam, and have a solid proof for one of the problems, but it's reliant on a number of theorems left exercises in the textbook which were not assigned as coursework. What, if any ...
4
votes
1answer
184 views

Found a simpler proof, now how do I know if it's original?

I've found a simpler proof for some identity/theorem, hypothetically speaking, of course ;) How do I know if it hasn't been done before? For important results it's fairly easy to find. By the way, I ...
3
votes
2answers
227 views

Introduction to proofs with a fair amount of hand-holding?

Lately I've gotten a friend of mine interested in mathematics. He has no college-level education to speak of, but is well employed as a software engineer. So I feel he's competent to learn this stuff ...
2
votes
2answers
180 views

How formal or informal should math texts (written for different purposes) be?

When writing math articles (or just math text), do you write down mathematical expression in a formal way or describe it in words, e. g. "Let $X$ be a normed vector space. Then $X$ is called a ...
8
votes
2answers
228 views

I feel the need to prove every result for myself

I am, at best, a novice mathematician. I started teaching myself the subject while writing my thesis in computer science. I find that I have a strong urge to prove every relationship or formula that I ...
18
votes
6answers
2k views

What are some common proof strategies in mathematics?

I want to start out by saying that I am new at proof based mathematics. I am used to seeing patterns and using them to solve similar problems. However, I have found this is not a very good way to ...
23
votes
4answers
927 views

Are there any common practices in mathematics to guard against mistakes?

It occurred to me that math is somewhat like programming (or vice-versa, if you prefer) because, in both, it is easy to make mistakes or overlook them, and the smallest error or misguided assumption ...