7
votes
3answers
94 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 ...
6
votes
4answers
656 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
113 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
89 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
59 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
68 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
119 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
632 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 ...
2
votes
2answers
141 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
272 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 ...
65
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
102 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
62 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
343 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
128 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
309 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
137 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
105 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
377 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
38 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
292 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 ...
24
votes
5answers
903 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
376 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
190 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
691 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
142 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
218 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?
4
votes
0answers
59 views

What do you do to facilitate proof clarity? [closed]

I tend to write proofs in the style of a paragraph or essay, using words (if-then) for the "logic" and reserving symbols (e.g. $\in, \subseteq$ etc.) for the "mathematics." (On the other hand, within ...
8
votes
4answers
403 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
604 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
667 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
99 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
182 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
224 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
179 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
225 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
919 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 ...
6
votes
6answers
1k views

Looking for Proofs Of Basic Properties Of Real Numbers

I have just begun my study of complex numbers and I learned where imaginary numbers came from and their importance. However there's one thing that I need to clarify and that is the properties of real ...
91
votes
9answers
4k views

Why do people use “it is easy to prove”?

Math is not generally what I am doing, but I have to read some literature and articles in dynamic systems and complexity theory. What I noticed is that authors tend to use (quite frequently) the ...
0
votes
2answers
103 views

how can I present an idea about the difference between two functions clearly?

I have a presentation in which I want to point out a difference between two functions. Instead of putting the two functions on a slide and pointing to the differences, I want to do something simple. ...
18
votes
6answers
805 views

Difficulty in Mathematical Writing

Lots of people (including myself) face lot of problems in tackling Mathematical Problems, which appear as if we can solve it, but then writing out a solution becomes difficult. Let us consider some ...