Questions tagged [proof-writing]

For questions about the formulation of a proof. This tag should not be the only tag for a question and should not be used to ask for a proof of a statement.

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377 votes
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Can every proof by contradiction also be shown without contradiction?

Are there some proofs that can only be shown by contradiction or can everything that can be shown by contradiction also be shown without contradiction? What are the advantages/disadvantages of proving ...
sonicboom's user avatar
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176 votes
9 answers
17k 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 ...
115 votes
12 answers
13k 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 ...
100 votes
11 answers
15k views

Does notation ever become "easier"?

I'm in my first semester of college going for a math major and it's pretty great. I'm doing well, however, there seems to be huge gap between how difficult /complex an idea is and how convoluted it is ...
Jonathan's user avatar
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79 votes
5 answers
63k views

Use of "without loss of generality"

Why do we use "without loss of generality" when writing proofs? Is it necessary or convention? What "synonym" can be used?
Pedro's user avatar
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78 votes
9 answers
6k views

Is it bad form to write mysterious proofs without explaining what one intends to do? [closed]

Often when doing assignments, I find myself deliberately writing in a "mysterious" way. By this I mean that the reader usually will not understand what exactly is going on and what for, until the very ...
Jaood's user avatar
  • 1,391
76 votes
8 answers
134k views

How do I prove that a function is well defined?

How do you in general prove that a function is well-defined? $$f:X\to Y:x\mapsto f(x)$$ I learned that I need to prove that every point has exactly one image. Does that mean that I need to prove the ...
Kasper's user avatar
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72 votes
3 answers
120k views

Prove: If a sequence converges, then every subsequence converges to the same limit.

I need some help understanding this proof: Prove: If a sequence converges, then every subsequence converges to the same limit. Proof: Let $s_{n_k}$ denote a subsequence of $s_n$. Note that $n_k \...
CodeKingPlusPlus's user avatar
71 votes
12 answers
7k views

Will assuming the existence of a solution ever lead to a contradiction?

I'm reading Manfredo Do Carmo's differential geometry book. In section 1-7, he discusses the "Isoperimetric Inequality" which is related to the question of what 2-dimensional shape maximizes the ...
Geoffrey's user avatar
  • 2,382
71 votes
11 answers
10k views

Is it technically incorrect to write proofs forward?

A question on an assignment was similar to prove: $$2a^2-7ab+2b^2 \geq -3ab.$$ and my proof was: $$2a^2-4ab+2b^2\geq0$$ $$a^2-2ab+b^2\geq0$$ $$(a-b)^2\geq0$$ which is true. However, my professor ...
mtheorylord's user avatar
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69 votes
6 answers
22k views

Strategies to denest nested radicals $\sqrt{a+b\sqrt{c}}$

I have recently read some passage about nested radicals, I'm deeply impressed by them. Simple nested radicals $\sqrt{2+\sqrt{2}}$,$\sqrt{3-2\sqrt{2}}$ which the later can be denested into $1-\sqrt{2}$....
JSCB's user avatar
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64 votes
8 answers
16k 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? [closed]

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 ...
Balbanna's user avatar
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64 votes
14 answers
6k views

What is a proof?

I am just a high school student, and I haven't seen much in mathematics (calculus and abstract algebra). Mathematics is a system of axioms which you choose yourself for a set of undefined entities, ...
user avatar
62 votes
7 answers
43k views

how to be good at proving? [duplicate]

I'm starting my Discrete Math class, and I was taught proving techniques such as proof by contradiction, contrapositive proof, proof by construction, direct proof, equivalence proof etc. I know how ...
uohzxela's user avatar
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61 votes
7 answers
6k views

Could I be using proof by contradiction too much?

Lately, I've developed a habit of proving almost everything by contradiction. Even for theorems for which direct proofs are the clear choice, I'd just start by writing "Assume not" then prove it ...
user64844's user avatar
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57 votes
2 answers
68k views

What exactly is the difference between weak and strong induction?

I am having trouble seeing the difference between weak and strong induction. There are a few examples in which we can see the difference, such as reaching the $k^{th}$ rung of a ladder and proving ...
user5826's user avatar
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53 votes
2 answers
2k views

On a long proof

On wikipedia there is a claim that the Abel–Ruffini theorem was nearly proved by Paolo Ruffini, and that his proof spanned over $500$ pages, is this really true? I don't really know much abstract ...
Ethan Splaver's user avatar
52 votes
3 answers
13k views

Where is the flaw in this "proof" of the Collatz Conjecture?

Edit I've highlighted the area in the proof where the mistake was made, for the benefit of anyone stumbling upon this in the future. It's the same mistake, made in two places: This has proven the ...
stevendesu's user avatar
50 votes
3 answers
14k views

How would one be able to prove mathematically that $1+1 = 2$?

Is it possible to prove that $1+1 = 2$? Or rather, how would one prove this algebraically or mathematically?
amizrahi's user avatar
  • 609
49 votes
3 answers
364k views

Limit of $(1+ x/n)^n$ when $n$ tends to infinity [duplicate]

Does anyone know the exact proof of this limit result? $$\lim_{n\to\infty} \left(1+\frac{x}{n}\right)^n = e^x$$
narendra-choudhary's user avatar
48 votes
8 answers
7k views

Systems of linear equations: Why does no one plug back in?

When someone wants to solve a system of linear equations like $$\begin{cases} 2x+y=0 \\ 3x+y=4 \end{cases}\,,$$ they might use this logic: $$\begin{align} \begin{cases} 2x+y=0 \\ 3x+y=4 \end{cases}...
Archr's user avatar
  • 1,141
47 votes
9 answers
262k views

What is the integral of 0?

I am trying to convince my friend that the integral of $0$ is $C$, where $C$ is an arbitrary constant. He can't seem to grasp this concept. Can you guys help me out here? He keeps saying it is $0$.
HowardRoark's user avatar
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47 votes
2 answers
4k views

The 'Factorialth Root'

I was dealing with the following question, given by my friend: Let $\xi(x)=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{\cdots}}}}$ Define the series $X$ as $\xi(1),\xi(2),\xi(3),\dots$ Find $n$ for ...
DynamoBlaze's user avatar
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47 votes
7 answers
6k 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 ...
BusySignal's user avatar
46 votes
6 answers
5k views

Why do we write proofs "forward?"

I am aware that this might turn into a discussion, but I have a feeling this might have an answer (maybe something historical?) instead. I'm hoping that those with speculations keep it in the comments....
MathMathCookie's user avatar
46 votes
4 answers
10k views

What really is mathematical rigor? How can I be more rigorous?

I'm an undergraduate mathematics student who has received some constructive feedback from two instructors at the end of my exams. Namely, that I am a bit hand-wavey and not always very rigorous. While ...
A. Thomas Yerger's user avatar
46 votes
7 answers
4k 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 of ...
user5826's user avatar
  • 12k
43 votes
3 answers
90k views

What is the proof that covariance matrices are always semi-definite?

Suppose that we have two different discreet signal vectors of $N^\text{th}$ dimension, namely $\mathbf{x}[i]$ and $\mathbf{y}[i]$, each one having a total of $M$ set of samples/vectors. $\mathbf{x}[m]...
hkBattousai's user avatar
  • 4,543
42 votes
2 answers
14k views

How to efficiently use a calculator in a linear algebra exam, if allowed

We are allowed to use a calculator in our linear algebra exam. Luckily, my calculator can also do matrix calculations. Let's say there is a task like this: Calculate the rank of this matrix: $...
cnmesr's user avatar
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42 votes
5 answers
7k views

Why do proof authors use natural language sentences to write proofs?

I haven't read very many proofs. The majority of the ones that I've read, I've read in my first-year proofs textbook. Nevertheless, its first chapter expatiates on the proper use of English in ...
Hal's user avatar
  • 3,386
42 votes
1 answer
3k views

How do I make proofs with long formulae more readable without sacrificing clarity?

Question A lot of things I'm trying to prove just now are turning into "notational hell", which I think makes them very hard to read. I've tried to cut down on this by assuming my reader ...
Ten O'Four's user avatar
  • 1,056
41 votes
10 answers
9k views

Why don't Venn diagrams count as formal proofs?

Just curious. If the purpose of a proof is to inform and persuade, why don't Venn diagrams count? Is it just convention or is there a more, umm, formal reason haha. Thanks!
papercuts's user avatar
  • 1,873
41 votes
4 answers
59k views

Explain proof that any positive definite matrix is invertible

If an $n \times n$ symmetric A is positive definite, then all of its eigenvalues are positive, so $0$ is not an eigenvalue of $A$. Therefore, the system of equations $A\mathbf{x}=\mathbf{0}$ has no ...
mauna's user avatar
  • 3,530
41 votes
8 answers
6k views

Should a mathematical proof be 'convincing'?

I just read a description of what is a mathematical proof in my mathematical logic textbook, and I'm a bit puzzled by it. It goes like this: A mathematical proof is a finite sequence of mathematical ...
Stephen's user avatar
  • 3,672
41 votes
2 answers
151k views

Proof for triangle inequality for vectors

Generally,the length of the sum of two vectors is not equal to the sum of their lengths. To see this consider the vectors $u$ and $v$ as shown below. By considering $u$ and $v$ as two sides of a ...
alok's user avatar
  • 3,880
40 votes
7 answers
8k views

How to attack "if true, prove it; if not true, give a counterexample" question?

I am taking a basic analysis course. This is a general question that I often encounter in weekly homework. How should we start to attack this type of question: if the statement is true, prove it; if ...
Q.L.'s user avatar
  • 957
40 votes
12 answers
66k views

Are There Any Symbols for Contradictions?

Perhaps, this question has been answered already but I am not aware of any existing answer. Is there any international icon or symbol for showing Contradiction or reaching a contradiction in ...
Mikasa's user avatar
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40 votes
1 answer
2k views

The art of proof summarizing. Are there known rules, or is it a purely common sense matter?

When a proof is long and difficult, it can be really nice vis-à-vis the reader to give a summary or an outline of the deduction before beginning hard work. Are there known rules to give a good proof ...
user avatar
39 votes
3 answers
4k views

How can I answer this Putnam question more rigorously?

Given real numbers $a_0, a_1, ..., a_n$ such that $\dfrac {a_0}{1} + \dfrac {a_1}{2} + \cdots + \dfrac {a_n}{n+1}=0,$ prove that $a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n=0$ has at least one real ...
Ovi's user avatar
  • 23.7k
37 votes
5 answers
1k views

Teacher claims this proof for $\frac{\csc\theta-1}{\cot\theta}=\frac{\cot\theta}{\csc\theta+1}$ is wrong. Why?

My son's high school teacher says his solution to this proof is wrong because it is not "the right way" and that you have to "start with one side of the equation and prove it is equal ...
Ramblin Wreck's user avatar
37 votes
5 answers
110k views

A practical way to check if a matrix is positive-definite

Let $A$ be a symmetric $n\times n$ matrix. I found a method on the web to check if $A$ is positive definite: $A$ is positive-definite if all the diagonal entries are positive, and each diagonal ...
Hannah's user avatar
  • 461
37 votes
3 answers
2k 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 ...
Michael Albanese's user avatar
36 votes
4 answers
6k views

prove $\sqrt{a_n b_n}$ and $\frac{1}{2}(a_n+b_n)$ have same limit

I am given this problem: let $a\ge0$,$b\ge0$, and the sequences $a_n$ and $b_n$ are defined in this way: $a_0:=a$, $b_0:=b$ and $a_{n+1}:= \sqrt{a_nb_n}$ and $b_{n+1}:=\frac{1}{2}(a_n+b_n)$ for all $n\...
doniyor's user avatar
  • 3,680
35 votes
11 answers
6k views

Does one accepted false statement allow proving anything?

When I was studying, the mathematical analysis professor said something interesting when he was explaining the implication (logical operator), namely $(False \implies True) = True$. He said something ...
Spook's user avatar
  • 946
34 votes
5 answers
39k views

Why does drawing $\square$ mean the end of a proof?

To end a proof, I often write "as was to be shown" or "q.e.d". Both of these terms make sense to me as a reader. On the other hand, I feel a little strange to put down $\square$ although I saw it ...
roxrook's user avatar
  • 12.1k
34 votes
6 answers
5k views

When has one sufficiently mastered an area of mathematics?

This is a rather soft question regarding the mastery of various mathematical subjects, such as undergraduate subjects. In particular, say, when has one mastered undergraduate analysis? Is it ...
Anthony Peter's user avatar
33 votes
5 answers
4k views

Is it okay to reverse engineer proofs in homework questions?

In a linear algebra text book, one homework question I received was: Prove that $\mathbf{a \cdot b} = \frac{1}{4}(\|\mathbf{a + b}\|^2 - \|\mathbf{a - b}\|^2)$. Where $\mathbf{a}$ and $\mathbf{b}$...
user3002473's user avatar
  • 8,943
33 votes
1 answer
41k views

Transpose of block matrix

I'm attempting to prove that $$ \left[ \begin{array}{c c} A & B \\ C & D \\ \end{array} \right]^\top = \left[ \begin{array}{c c} A^\top & C^\top \\ B^\top & D^\top \\ \end{array} \...
Red's user avatar
  • 333
33 votes
2 answers
37k views

Prove that a set $E$ is closed iff it's complement $E^{c}$ is open

I was wondering if this proof was right. $\Leftarrow$ Suppose $E$ is closed. Then choose $x\in E^{c}$, then $x\notin E$, and so $x$ is not a limit point of $E$. Hence there exists a neighborhood $N$...
user77107's user avatar
  • 819
32 votes
6 answers
56k views

A subset of a compact set is compact?

Claim:Let $S\subset T\subset X$ where $X$ is a metric space. If $T$ is compact in $X$ then $S$ is also compact in $X$. Proof:Given that $T$ is compact in $X$ then any open cover of T, there is a ...
Mathematics's user avatar
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