For questions about the writing of proofs. But instead of focusing the mathematics behind the proof, these questions ask about the details and implementation of the proof.

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1
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1answer
36 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 ...
4
votes
0answers
48 views

Handwaving gone wrong

My motivation for this question is twofold: On one hand, I'm studying algebraic topology, where - at least in the book written by Hatcher - there is quite a lot of handwaving (e.g. maps are continous ...
0
votes
0answers
21 views

Subset of a finite set is finite: base step

We can prove by induction that any subset of a finite step is finite. But I am confused by the step "Observe first that all subsets of $\emptyset$ and $\mathbf I_1$ are finite", which I think is the ...
0
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2answers
30 views

Proving the arithmetic mean equals the geometric mean when $a=b$.

Arithmetic mean $a,b \in \mathbb R$ is $A(a,b)=\frac{a+b}{2}$ Geomtric mean $a,b \in\left[0,\infty\right]$ is $G(a,b)=\sqrt{ab}$ I'm trying to prove that $G(a,b)=A(a,b)$ if and only if $a=b$. ...
1
vote
1answer
40 views

Maximum and average number of inversions in array by induction

Just for your information, an inversion in an array $a$ is any ordered pair of points $(i, j)$ where $i < j$ and $a_i > a_j$. I can prove the maximum and average number of inversions in an ...
0
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2answers
36 views

The maximum no. of edges in a DISCONNECTED simple graph…

... on n vertices when it is not connected being equal to (1/2)(n - 1)(n - 2)... I can see that for n = 1 & n = 2 that the graphs have no edges... however I don't understand how to derive this ...
0
votes
1answer
52 views

Check my proof by contradiction…

The question is to prove the following by contradiction. There does not exist a smallest positive non-zero rational number. What I tried... There does exist a smallest positive non-zero rational ...
2
votes
6answers
129 views

Proving that $\int \frac{1}{x} \mathrm dx = \ln(|x|) + C_1$

In all textbooks and online notes, there is always a table of antiderivatives and it always says $\int \frac {1}{x} \mathrm dx = \ln(|x|)+C_1$ but there is nowhere a proof. I found some proofs online ...
3
votes
3answers
58 views

Group of even order must contain $a:a=a^{-1}$ $ (a\not = e)$ [duplicate]

Let $G$ be a finite group. If the order of $G$ is even, prove that there is at least one element $a$ in $G$ such that $a\not= e$ and $a=a^{-1}$. Here's my idea: Suppose $\{x_1,\cdots,x_n\}$ is ...
2
votes
1answer
34 views

Vector spaces - Multiplying by zero vector yields zero vector.

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is solely based on vector space axioms. Axiom ...
1
vote
0answers
36 views

Vector spaces - Multiplying by $-1$ yields inverse element of vector addition.

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is based on vector space related axioms. Axiom ...
0
votes
1answer
19 views

How to show a triple represents all possible selections?

Let $Y=\{y_1, y_2, y_3,y_4,y_5\}$ Then, the choices of selecting 3 objects (repetitions allowed) from $Y$ can be represented by the triple $(y_{i_1},y_{i_2},y_{i_3})$ where $i_1 \le i_2 \le i_3$. Is ...
0
votes
1answer
41 views

How to prove a function from a set of triples to $\mathcal{P}_3(\mathbb{N}_7)$ is a bijection?

Let $Y=\{y_1, y_2, y_3, y_4,y_5\}$ The function from the set of triples $(y_{i_1},y_{i_2},y_{i_3})$ where $i_1 \le i_2 \le i_3$ to $\mathcal{P}_3(\mathbb{N}_7)$ is a bijection given by ...
0
votes
0answers
15 views

proof of log-sum giving maximum value given equality constraint

How to prove the following equation: $$ -\log\sum_{k=1}^K f_k=\min_{\bf{u}}-\sum_{k=1}^K u_k \log(f_k) +\sum_{k=1}^K u_k \log(u_k)\\ s.t.\ u_k \in (0,1), \sum_k u_k=1 $$ using Lagrangian multiplier? ...
2
votes
1answer
60 views

Prove $ x^n-1=(x-1)(x^{n-1}+x^{n-2}+…+x+1)$

So what I am trying to prove is for any real number x and natural number n, prove $$x^n-1=(x-1)(x^{n-1}+x^{n-2}+...+x+1)$$ I think that to prove this I should use induction, however I am a bit stuck ...
1
vote
1answer
27 views

Induction proof for continued fractions

Recently while preparing for a maths test, I got this question in a book: Let $a(n) = 3 + \cfrac{1}{3+\cfrac{1}{3+\cfrac{1}{3+\cdots }}}$ till $n$ terms. Prove that $a(n) \cdot a(n-1)=3a(n-1)+1$ ...
-3
votes
0answers
30 views

Has anyone read this book on Proofs? [closed]

Seems really good and thorough. I'm about to go through it before I try to work through Spivak. I'm trying to build more rigor in Mathematics. "Bridge to Abstract Mathematics: Mathematical Proof and ...
2
votes
1answer
31 views

Writing up a rigorous solution for finding a basis for the $n \times n$ symmetric matrices.

I am aware that a similar question has been asked here, among other questions, but I feel that my question is different because I am actually trying to write up a very rigorous proof that such a set ...
2
votes
2answers
51 views

Writing solutions of inequalities: $3<x$ versus $x>3$

My son wrote a solution to a number line graph as 3 < x instead of what his teacher said was the correct answer of x > 3. When he brought his paper back in to bring it up he was told that the ...
0
votes
0answers
37 views

Proof Strategy: Induction Summation of Series

Let $P(n)$ be the following statement: $$\sum\limits^{n}_{i=0}r^i = \dfrac{1-r^{1+n}}{1-r}\text{ for all }n \in \mathbb{N}\text{.}$$ I am stuck at the base case: $$P(1):1 + r = ...
0
votes
1answer
26 views

Infinite Wilson Prime proof

An article I read recently about Wilson Primes stated that, while 5, 13, and 563 are the only known terms, there is an infinite number of Wilson Primes. I was wondering if someone could verify this ...
0
votes
1answer
25 views

Prove that for any $\{x_{m_k} \}\in I_n$, where $I_n$ are dyadic intervals, $\lim_{n \to \infty} x_{m_k} =c$

Proving for any $\{x_{m_k} \}\in I_n$ that:$$\lim_{n \to \infty}\{x_{m_k} \}=c$$ I have been trying to solve this problem, but i dont know how to write it properly, so i need your help whit ...
0
votes
1answer
25 views

Help for understanding Danielson-Lanczos lemma

The Danielson-Lanczos lemma is the basis for fast Fourier transform algorithms. Now, I do understand this step $\displaystyle X_{k} = \sum_{n=0}^{N-1} x_{n}\omega^{kn}_{N} = \sum_{n=0}^{(N/2)-1} ...
0
votes
2answers
40 views

Formal Proof: P(A∩B'∩C') = P(A) - P(A∩B) - P(A∩C) + P(A∩B∩C)

I'm trying to prove the following: $\newcommand{\P}{\operatorname{\bf P}}\P(A\cap \overline{B}\cap\overline{C}) = \P(A) - \P(A\cap B) - \P(A\cap C) + \P(A\cap B\cap C)$ I can explain it with a venn ...
1
vote
3answers
51 views

Formal proof of $\lim_{x\to a}f(x) = \lim_{h\to 0} f(a+h)$

$$ \lim_{x\to a} f(x) = \lim_{h\to 0} f(a+h) $$ How do I write a formal proof of it?
0
votes
0answers
20 views

Proof that the $sqrt[k]{z}\, z \in \mathbb N$ counts the amount of numbers less than or equal to z with a $k-$exact power

Empirically, it can shown that $$\mathrm{Floor}[\sqrt[k]{z} ] \,, z \wedge k\in \mathbb N $$ is equal to the amount of numbers which have a $k-$exact root. For example, $\sqrt 36 = 6$ means that there ...
2
votes
0answers
43 views

How to model a real-world graphical structure into a mathematical formulation?

I am trying to learn how to model programming problems in a mathematical way. I am a software engineer, but have recently been running into road blocks where I can't solve some problems very ...
2
votes
0answers
34 views

Proof of Gersgorin Discs Theorem

I have a question about the proof of Gersgorin Theorem from the book Matrix Analysis by Horn & Johnson. The Theorem states that for any $A\in \mathbb{C}^{n \times n}$ 1) all eigenvalues are ...
0
votes
2answers
159 views

Help showing that every walk of length $k$ from $x$ to $y$ in a graph is a path.

If I were to suppose $x$ and $y$ are two vertices in the same connected component of a graph, and let $k$ be the distance between them, how would I prove that every walk of length $k$ from $x$ to $y$ ...
0
votes
1answer
26 views

How to handle a function from a set of functions to another set of functions?

Given sets $X$ and $Y$ we denote the set of functions from $X$ to $Y$ by $\text{Fun}(X,Y)$. Let: $k,n \in \mathbb{Z}^+$ $X_1 = \{x_1,x_2\dots, x_{k+1}\}$ $Y = \{y_1, y_2, \dots , y_n\}$ Then, ...
3
votes
0answers
58 views

Vector spaces - Multiplying by zero scalar yields zero vector

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is solely based on vector space related axioms. ...
4
votes
1answer
44 views

Vector spaces - If an addend adds nothing, then the addend is the zero vector.

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. With one exception, the following proof is solely based on vector ...
1
vote
0answers
25 views

Another proof of uniqueness of identity element of addition of vector space

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is solely based on vector space axioms. Axiom ...
4
votes
4answers
79 views

Construction of an equilateral triangle from two equilateral triangles with a shared vertex

Problem Given that $\triangle ABC$ and $\triangle CDE$ are both equilateral triangles. Connect $AE$, $BE$ to get segments, take the midpoint of $BE$ as $O$, connect $AO$ and extend $AO$ to $F$ where ...
3
votes
5answers
305 views

Proof of uniqueness of identity element of addition of vector space

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. Proof of uniqueness of identity element of addition of vector space ...
0
votes
3answers
39 views

How to write $a|b \wedge ((a=0 \wedge b=0) \vee (a\ne0 \wedge b\ne0))$

How can I write the expression $$a|b \wedge ((a=0 \wedge b=0) \vee (a\ne0 \wedge b\ne0))$$ concisely and clearly in English? A direct translation yields $a$ divides $b$ and either {$a$ and $b$ ...
1
vote
1answer
25 views

To prove these sets are equal without using modulo arithmetic.

Prove $\{3t : t \in \mathbb Z\} \cup \{3t + 1 : t \in \mathbb Z\} \cup \{3t + 2 : t\in \mathbb Z\} = \mathbb Z.$
1
vote
1answer
62 views

Odd town Even town explanation.

I am struggling to understand the solution to the following problem: If $\mathcal F\subset 2^{[n]}$ such that for each $F_1$ and $F_2$ in $\mathcal F$ we have $|F_1|,|F_2|\equiv 1 \bmod 2$ and ...
0
votes
1answer
33 views

How prove that $|QA| < |QC|$ in triangle?

$ABC$ is a triangle with a right angle at $A$, and $|AB|$ > $|AC|$. The point $D$ is defined so that $BCD$ is equlateral and $AD$ intersects $BC$ at $P$. The point $Q$ is defined so that $QDP$ is ...
1
vote
1answer
33 views

For what k is $\mathcal{M}_{m \times n}$ isomorphic to $\mathbb{R}^{k}$?

For what k is $\mathcal{M}_{m \times n}$ isomorphic to $\mathbb{R}^{k}$ ? I get a feel but am unable to prove it.
4
votes
1answer
77 views

How show that $ABC$ is equilateral?

Let $D$, $E$ and $F$ be three points on sides $BC$,$AC$ and $AB$ of triangle $ABC$ such that lines $AD$, $BE$ and $CF$ concur at point $M$. If three trianles $MDB$, $MCE$ and $MAF$ have equal areas ...
3
votes
1answer
17 views

Proof About Point and Triangles

Suppose we are given $n$ points in a plane, where $n\ge 4$ and no 3 of the points are collinear. If $k$ distinct triangles are designated with vertices among the $n$ points, show that no more than ...
1
vote
1answer
43 views

How prove that $AD>BE$ in triangle?

Let $D$ be a point on the side $BC$ of a triangle $ABC$ such that $AD>BC$ . The point $E$ on $CA$ is defined by the equation $\frac{AE}{EC}=\frac{BD}{AD-BC}$ .How prove that $AD>BE$?
0
votes
1answer
31 views

How to prove $|q|\ge 1 \Rightarrow |a|\ge |d|$?

Let $a,d,q \in \mathbb{Z}$ and $a=dq$ How do I show that $|q| \ge 1 \Rightarrow |a| \ge |d|$? I've tried: $|q|\ge 1 \Rightarrow (q>1 \text{, if } q>0) \text { or } (-q>1 \text{, if } ...
0
votes
2answers
52 views

Prove intersection of open balls is another open ball

I was wondering how I would prove that an intersection of two open balls is also another open ball. The definition I have of an open ball is: If x $\in X$ and $\epsilon > 0$, $B_{\epsilon}(x) :=$ ...
-3
votes
1answer
59 views

State whether the following claim is true, and then prove or disprove it. Give a detailed structured proof, justifying every step. [closed]

State whether the following claim is true, and then prove or disprove it. Give a detailed structured proof, justifying every step. $$\forall n \in \mathbb{N}, [(\exists k \in \mathbb{N}, n = ...
2
votes
4answers
69 views

Is this a valid proof for the existence of a rational number between any two real numbers?

Given $a, b \in \mathbb R$ with $a<b$, prove that there exists some $r \in \mathbb Q$ such that $a<r<b$. Before I prove the main statement, there's a lemma I'd like to prove: Lemma ...
0
votes
2answers
34 views

Trying to disprove a statement - some partial working included

I am trying to find a counter example to show that the statement below is false, but I am having difficulty in trying to find a reasonable argument. Here is the statement: $n^2-12n + 35 \geq 0$ for ...
0
votes
3answers
83 views

Prove $x^2=t$ for any $t>0$ [duplicate]

Prove for any positive number $t$, there is a solution for $x^2=t$. So we want to show that $x^2=t$ for $t\geq0$. We can break this into two cases: Case 1: Assume $t=0$, then we have $x^2=0$ ...
2
votes
2answers
50 views

Surjectivity of a piecewise function $f:(-1,1)\to \mathbb R$

Function $f$ is defined as $f: (-1, 1) \to \mathbb{R}$. $$ f(x) = \begin{cases} -x/(x-1),&x\geq 0 \\ x/(x+1),&x \leq 0 \end{cases} $$ Let $y \in \mathbb{R}$. How would I prove that there ...