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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Help with proof in complex analysis

I was looking at the proof of the result, the image of $\mathbb R_\infty$ under mobius transformation is a circle. I don't follow how does this step $(a\bar d-\bar bc)w+(b\bar c-\bar ad)\bar w+b\bar ...
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1answer
10 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 ...
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1answer
33 views

My first proof related to subspaces (vector spaces). Please comment.

What do you think about my first proof which deals with subspaces? Theorem An intersection of subspaces is a subspace. Preliminaries Corresponding to the notation in Wikipedia, symbols for vectors ...
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4answers
31 views

Induction Proof: $\sum_{k=1}^{n}\frac{a-1}{a^k}=1-\frac{1}{a^n}, a \ne 0$

I'm having trouble showing equality for the $A(n+1)$ portion of the proof. Prove by Induction: $$\sum_{k=1}^{n}\frac{a-1}{a^k}=1-\frac{1}{a^n}, a \ne 0$$ Base Case $(n=1)$: ...
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2answers
108 views

Proof of $x*0=0$ is invalid?

My professor told me today that while my logic was good and all my steps after the assumption were correct, my argument was invalid because I was assuming what I want to prove. I don't see how. ...
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2answers
54 views

Checking understanding on proving uniqueness of identity and inverse elements of a group.

Sorry for such a trivial question, but just wanted to check my understanding. When proving a statement, for example, that the inverse of a group element is unique (in elementary group theory) one ...
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3answers
49 views

How to prove that a set is infinite iff it is Dedekind infinite?

I need to prove the following: A set $X$ is infinite if and only if it is equipotent to a proper subset of itself Here, $X$ is defined to be infinite if $|X|$ is not a non-negative integer or ...
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2answers
77 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 ...
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16answers
1k views

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

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 ...
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2answers
24 views

Proving injective (1-1) using contrapositive

Given function $f:\mathbb Z \to \mathbb Z$ defined by $f(n) = n - 6$ $\mathbb Z$ in this case is the set of integers. Suppose for $x_1$, $x_2 \in \mathbb Z$, we have $f(x_1) = f(x_2)$. This means ...
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1answer
134 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. ...
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1answer
74 views

Verify my proof: If $X$ is infinite, then there exists $f: \mathbb{N} \rightarrow X$ such that $f$ is injective.

Proposition: If $ X $ is infinite, then there exists $ f: \mathbb{N} \rightarrow X $ such that $f$ is injective. Proof: Define $X$ as a infinite set, i. e., there does not exist $ g: [k] \rightarrow ...
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1answer
45 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 ...
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5answers
89 views

How would I show that R is an equivalence relation?

If I were to consider the relation R on ℤ defined by n R m if and only if P(n)=P(m). How would I show that R is an equivalence relation? Any help is appreciated.
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3answers
48 views

Direct proof and contradicttion

If a statement has direct proof, it can not be proved by contradiction. Is it true? I want to know if a question has direct proof: can we once again prove by contradiction method too?
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2answers
89 views

How to show that an infinite decimal is equal to a unique real number?

I don't understand how the proof above shows that two distinct real numbers correspond to different infinite decimal. All I got out of the explanation is given any two distinct real numbers $a$ and ...
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3answers
46 views

How to conclude $|a|<|b|$ from $a<\frac{b^2}{a} \text{ and } \frac{a^2}{b}<b$? (Direct Proof)

The original question is to prove that for all real numbers $a$ and $b$, $a^2 < b^2 \Rightarrow |a| < |b|$. I was able to easily prove this by proving that its contrapositive, $|a|\ge|b| ...
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5answers
158 views

The set of all finite subsets of the natural numbers is countable

Could someone verify my proofs? Proposition: the set of all finite subsets of $\mathbb{N}$ is countable Proof 1: Define a set $ X=\{A\subseteq\mathbb{N}\mid \text{$A$ is finite} \}$. We can have a ...
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3answers
59 views

Explanation for the number of partitions of $\{1,\dots,n\}$ into $k$ parts

A partition of the set $\{1, 2, . . . , n\}$ into $k$ parts is a way of writing the set as a disjoint union of $k$ subsets. For example $\{1, 2, 3, 4, 5\} = \{1, 4\} \cup\{2, 3\} \cup \{5\}$ is a ...
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0answers
35 views

IS this proof by induction of the hand shake lemma correct?

Proof by induction that the sum of degrees of vertexes in an undirected graph equals two times the number edges, where $V$ is the set of vertexes and $E$ is an edge multiset: $$\sum_{v ∈ V} deg(v) = ...
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3answers
75 views

Directly prove that $2x^2 -4x + 3 > 0$ for all real $x$

I'm asked (for homework which isn't graded but instead the basis of a quiz) to directly prove that $2x^2 -4x + 3 > 0$ for all real $x$. I am VERY new to proofs. The textbook's only example is a ...
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1answer
43 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 ...
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2answers
91 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 ...
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0answers
24 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 ...
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2answers
34 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 $a=b$. ...
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1answer
42 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 ...
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2answers
40 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
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1answer
54 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
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6answers
136 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
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3answers
63 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
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1answer
38 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 ...
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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 ...
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1answer
20 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 ...
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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 ...
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0answers
18 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
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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 ...
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1answer
29 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 $a(n) \cdot a(n-1)=3a(n-1)+1$ ...
2
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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
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2answers
52 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 ...
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0answers
38 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 = ...
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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 ...
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1answer
28 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 ...
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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} ...
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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 ...
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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?
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0answers
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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
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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
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0answers
35 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 ...
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2answers
160 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
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1answer
29 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, ...