For questions about the formulation of a proof, not about the mathematics behind it.

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0answers
35 views

A set A is infinite if and only if there is a bijection from the set A to a proper subset of A. [duplicate]

I'm just starting my journey into proof writing and I don't really know how to do this. More specifically I think I want a proof of the fact that every infinite set A is Dedekind-infinite (i.e. that ...
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2answers
37 views

Implication vs Equivalence in proofs

I understand the definition of both the implication and equivalence signs. When I get asked to prove something, I will probably have to use both implication and equivalence logic. My question is if it'...
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4answers
140 views

Proof of greatest common divisor [duplicate]

The greatest common divisor of two positive integers $a$ and $b$ is the largest positive integer that divides both $a$ and $b$ (written $\gcd(a, b)$). For example, $\gcd(4, 6) = 2$ and $\gcd(5, 6) = 1....
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1answer
34 views

Simple examples of applications of converse, contrapositive and inverse used in mathematical proofs rather than logic.

While learning simple logic in high school, I remember learning about converse, contrapositive and inverse (maybe some others as well). Yet, I don't seem to recall their usage for proofs (only ...
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1answer
40 views

Proof via mean value theorem

Suppose $f$ is differentiable with $f(0)=0$; $f'(x)<0$ for all $x<0$ and $f'(x)>0$ for all $x>0$. Prove $f(x)\geq 0$ for all $x\in\mathbb{R}$. It's pretty clear to me that $f$ decreases ...
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1answer
34 views

How to prove ~ is an equivalence relation

In a metric space $M$, declare $x \sim y$ to mean that there is a continuous function $\gamma : [0, 1] \rightarrow M$ such that $\gamma(0) = x$ and $\gamma(1) = y$. Prove that $\sim$ is an equivalence ...
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1answer
28 views

Proof: An accumulation point if each neighborhood contains point not itself

I want to confirm my proof of this lemma. Lemma: Let $S$ be a set of real numbers. Then $a$ is an accumulation point of $S$ if and only if there each neighborhood of $a$ contains a member other than ...
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3answers
52 views

$xf'(x) = αf(x)$. How to prove that $f(x) = cx^\alpha$?

Let $f$ be a differentiable function such that $xf'(x) = \alpha f(x)$ for all $x > 0$. How do I show that $f(x) = cx^\alpha$ for some constant $c$? I have $f'(x) = \alpha f(x)/x$ , and I can see ...
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2answers
89 views

Proving $n\sin(\frac{\pi}{n})<\pi<n\tan(\frac{\pi}{n})$ ; obtaining results from it.

I was reading The Simpsons and the Mathematical Secrets when I encountered the story of $\pi$. It mentions how Archimedes devised a method to place a lower and upper bound on $\pi$ by bounding a ...
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1answer
94 views

Question about this specific proof of $\forall a\in G$ $aH=Ha$ implies $H$ is normal

I've just tried to come up with a proof of the above statement but I feel like something is not quite right. The question I have isn't about proving the statement, I've found that in lots of places, ...
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2answers
20 views

Simplest Proof for an Elementary Number Theory Condition

I have to proove in the most 'primitive' way the following: if $ab=ac$ then either $a=0$ or $b=c$. I could think only about the following solution: considering the given $ab=ac$, let's subtract from ...
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1answer
67 views

Let A be a denumerable set. Prove that the set $\{B:B\subset A\}$ and cardinality of B=1 of all 1-element subsets of A is denumerable.

So my original idea was to show that the countable union of countable sets is countable since I know that each set has one element. I'm not exactly sure how to start this off though. Thanks in advance....
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2answers
59 views

Prove or disprove: If $f(x): \mathbb{R} \rightarrow \mathbb{R}$ is bounded above, then $f(x+10)$ is also bounded above.

We want to prove or disprove: If $f(x): \mathbb{R} \rightarrow \mathbb{R}$ is bounded above, then $f(x+10)$ is also bounded above. 1) I need to first identify if this statement is true or not true. ...
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1answer
76 views

If $\{B_{i} : i\in N\}$ is a denumerable family of pairwise disjoint distinct finite sets, then $\bigcup_{i\in N} B_{i}$ is denumerable.

So for this proof I'm thinking of creating a bijection from N to $\bigcup_{i\in N} B_{i}$ , but I'm not sure how I should go about doing this. I thought about listing out a couple of sets from $\{B_{...
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1answer
131 views

Graph theory: proving that a graph with specific property is bipartite

I have been given the following problem on an exercise sheet: Let $G$ be a graph with $n$ vertices with the property that for each $k ≤ n$, every set of $k$ vertices contains a subset of size at ...
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2answers
45 views

Common methods for proving: Existence proof

For every real number x with $x\neq -1$ there exists a real number y such that $ \frac{y}{y+1}=x $. $ \forall x\in \Bbb R\setminus (1) \ \ \exists y \in \Bbb R : \frac{y}{y+1}=x \\, x \neq -1 $ $ ...
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1answer
33 views

Proof strategy about a property of triangular matrices

Is it by mathematical induction the best way to prove that the determinant of an upper (lower) triangular matrix is the product of the elements of the main diagonal? Actually, I am wondering about ...
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1answer
43 views

Let $f(x) = x^2 - 4x - 12$ prove that $f$ is not uniformly continuous on the set $[-2,∞)$

so I know that to show this is NOT uniformly continuous then I need to show that $\exists \epsilon>0$ $\forall \delta>0 \exists x,y\in [-2,\infty)$ such that $(|x-y|<\delta ~~\&~~ |f(x) - ...
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1answer
46 views

Proving the area of a triangle within a triangle

Consider a triangle with vertices ABC, we pick a point C' on the line segment AB in such a way that |BC'|=2|AC'|. Similarly, we pick a point B' on the line segment AC and a point A' on the line ...
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2answers
113 views

Prove that $\lambda = 0$ is an eigenvalue if and only if A is singular, without using $\lambda_1\cdot\ldots\cdot\lambda_n = det(A)$. [duplicate]

I would like to know if there is any proof without using the fact that: $$\lambda_1\cdot\lambda_2\cdot\ldots\cdot\lambda_{n-1}\cdot\lambda_n = det(A)$$ I managed to prove that if $\lambda = 0$ then, ...
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1answer
33 views

Prove that minimum spanning tree is a tree

From the the Wikipedia page Minimum spanning tree: A minimum spanning tree is a spanning tree of a connected, undirected graph. It connects all the vertices together with the minimal total ...
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2answers
51 views

Prove that $\{w \mid \text{ w has even length and the first half of w has more 0s than the second half of w} \}$ is not regular?

I have had some difficulties understanding proofs that a language is not regular using the Pumping Lemma, and now I need to prove that the following language $$A = \{w \mid \text{ w has even length ...
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1answer
34 views

Show that if $S$ is an orthogonal set of nonzero vectors in $\mathbb{R}^n$ then $S$ is a linearly independent subset of $\mathbb{R}^n$.

Use the dot product to show that if $S = \{\vec{v_1}, \vec{v_2}, ..., > \vec{v_n}\}$ is an orthogonal set of nonzero vectors in $\mathbb{R}^n$ then $S$ is a linearly independent subset of $\...
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0answers
52 views

Find and prove $\lim_{n\to\infty}\sum_{k=0}^n x^k$

Find: $$\lim_{n\to\infty}\sum_{k=0}^n x^k$$ Where x is a set of the complex numbers and $|x|<1$ Then prove the limit using $\epsilon$ and $N_0(\epsilon)$ I think the limit is either $0$ or $\...
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2answers
29 views

Proving or disproving statements about operations with integers

I'm really stuck with this one and I'm thankful for any help. Consider the following operations on the set of integers: $\hspace{8em} a\star b := a^2 + b^2 \hspace{5em} a\diamond b := a+b+2ab$ ...
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0answers
36 views

Distinctions of different topologies on the sequence space (countable cartesian products of $\mathbb{R}$)

$\newcommand{\b}[1]{\mathbf{#1}}$ $\newcommand{\R}{\mathbb{R}}$ $\newcommand{\N}{\mathbb{N}}$ Question I solved this exercise in Munkres.(20.4) But I don't know if I did it righ t or not. I really ...
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0answers
20 views
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1answer
53 views

Proof involving an $\iff$ statement related to eigenvalues and eigenvectors

I have the following theoreom: Let us consider the matrix $A\in\mathbb{R}^{n,n}$. $\lambda$ is an eigenvalue of $A$ iff $(\lambda I - A)$ is not invertible, or, equivalently, $\det (\lambda I - A) ...
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2answers
38 views

Square of Elementary Matrix Proof

I'm having trouble proving the following statement: "There exists an elementary matrix $E_1$ such that $E_1^2 = I$" I'm thinking about how the inverse of $E_1$ is equal to $E_1$ (so $E_1^{-1} = E_1$)...
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1answer
23 views

Proof regarding the convergent series of a subsequence

Given a null sequence $(a_n)_{n\in\mathbb{N}}$ and $a_n\in\mathbb{R}^{>0},\forall n\in\mathbb{N}$ I need to prove that $\forall\epsilon >0,\exists (a_{n_k})_{k\in\mathbb{N}}$ such that $(\sum_{...
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3answers
370 views

Prove that $\ln(x)=\frac{1}{x}$ has a unique solution

The question is like: Prove that the equation $\ln(x)=\frac{1}{x}$ for $x>0$ has a unique solution and explain why. When it asks about the "unique solution" I try to find the exact value. Is ...
2
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1answer
212 views

Proof/intuition that any number can be expressed in binary form and every number will have a unique representation?

I was just thinking lately that how do we know that literally every number can be expressed in binary? And that too, with a unique representation? Clarification: With numbers, I mean whole numbers. ...
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1answer
18 views

Perpendicular (orthogonal) vector proof

Show that two nonzero vectors $\vec{v_1}$,$\vec{v_2}$ ∈ $\mathbb{R_3}$ are orthogonal if and only if their direction angles satisfy cos$α_1$ cos$α_2$ +cos$β_1$ cos$β_2$ +cos$γ_1$ cos$γ_2$ =0. Note: I ...
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3answers
90 views

Use Cauchy-Schwarz Inequality to prove statement

Use the Cauchy-Schwarz inequality to show that (acos(θ)+bsin(θ))$^2$ $\leq$ a$^2$ +b$^2$ for all a,b,θ ∈ $\mathbb{R}$ What I was trying to do was to take the smaller of either a or b and prove that (...
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0answers
11 views

Computing a sum of functions of $\xi$-adhering numbers: $\big\lfloor{\xi \over\lfloor{\xi/\beta}\rfloor}\big\rfloor=\beta$

Assume the existence of a $\xi \in Z^+$. A $\beta \in Z^+ ; 1\le\beta\le \xi$ is said to be $\xi$-adhering if $$\big\lfloor{\xi \over\lfloor{\xi/\beta}\rfloor}\big\rfloor=\beta$$ Let $f(\xi)$ be ...
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9answers
2k views

Prove that for every three non-zero integers, a,b and c, at least one of the three products ab,ac,bc is positive

The question is: Prove that for every three non-zero integers, a,b and c, at least one of the three products ab,ac,bc is positive. Use proof by contradiction. My general approach to doing ...
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1answer
34 views

How do you show that the estimator for the covariance matrix is unbiased?

So according to Wikipedia (Here) the sample covariance matrix is an unbiased estimator of the covariance matrix, but how do I prove this mathematically?
3
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1answer
128 views

Prove a complex number is real

Let $z$ be a complex such that $|z-1| =1$, and consider the complex numbers $v$ and $w$ such that: $w = z^2 -z$ and $3\arg(v) = 2\arg(w)$, where arg is the argument of a complex number. Show that $$ \...
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1answer
30 views

Proving a group homomorphism: $\phi: G\rightarrow \text{Aut}(G), g\mapsto g^*$

Given $\phi: G\rightarrow Aut(G), g\mapsto g^*$ and $g^*: G\rightarrow G, x\mapsto gxg^{-1}$ where $g,g^{-1},x\in G$ I need to prove that $\phi$ is a homomorphism ($\phi(gh)=\phi(g)\phi(h)$). So, ...
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1answer
84 views

Proof that the exponential function is continuous on $\mathbb{R}$ without use of derivatives

I am still trying to understand how to prove statements. I want to prove that for $a>0$, $f(x) = a^x$ is continuous on $\mathbb{R}$. The text gives an hint, namely, that it suffices proving ...
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2answers
57 views

Proof involving Floor function

Prove: $\forall l \in \mathbb{Z}: \forall r \in \mathbb{R}: 0 \le r \lt 1 \Rightarrow\lfloor l+r\rfloor = l$ My attempt: Let: $l \in \mathbb{Z}, r \in \mathbb{R}.$ Assume: $0 \le r \lt 1.$ $\...
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3answers
47 views

what is the difference between ∀x∈ℝ, ∀ε>0, (|x|≤ε ⇒ x=0) and ∀x∈ℝ, ((∀ε>0, |x|≤ε) ⇒ x=0)

I am supposed to Prove or disprove: ∀x∈ℝ, ∀ε>0, (|x|≤ε ⇒ x=0) and: ∀x∈ℝ, ((∀ε>0,|x|≤ε) ⇒ x=0) I think I understand how to prove the second one (by contradiction) but I don't understand what makes it ...
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1answer
160 views

For all real numbers x, if x −⌊ x ⌋≥ 1/2 then ⌊2x ⌋= 2⌊x ⌋+ 1.

I had this problem as a homework assignment and had to write a proof for it. I've tried some approaches but keep getting stuck. This is what I have so far: Suppose that x is any particular but ...
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0answers
25 views

Subgroup of the permutation group of $G$. Did I prove this right?

Given a group $G$, the automorphism group $Aut(G)$ of $G$ and the permutation group $\sum_G$ of $G$, I have to prove, that $Aut(G)$ is a subgroup of $\sum_G$. To do that I must prove: $Aut(G)$ has ...
0
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1answer
19 views

Proving a sequence in a compact set contains a point that converges to zero

Let $E$ be a compact set and let $f$ be continuous on $E$. Suppose there is a sequence {$x_n$}$_{n=1 \to \infty}$ of such points of E such that $\lvert f(x_n) \rvert < \frac{1}{n}$ for each $n$. ...
0
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2answers
56 views

Proving that a compact set which has a limit point at each point in the set is bounded

Suppose $f:K \to (-\infty, \infty), K $ is compact, and $f$ has a finite limit at each point of $K$, but may not be continuous on $K$. Show that f is bounded. Then what if we don't know if $f$ is ...
0
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1answer
29 views

Question with proof by contradiction

Given a collection of numbers, one may wish to find the "closest pair": two numbers in the collection that are not the same, but whose difference is as small as possible. For instance, if we have ...
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2answers
86 views

Prove this theorem: If the product of two consecutive integers is not divisible by 6, then it can be written in the form 9t+2 where t is an integer.

I know that product of two consecutive integers must be even, but not too sure how it helps in proving this.
2
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2answers
54 views

Proof (cases & induction): Find the set of positive integers such that $n! \geq n^3$

I need to find the set of positive integers such that $n! \geq n^3$, and then prove my answer is true using cases and induction on $n$. There is a lemma that I will need to prove and use for this ...
2
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1answer
71 views

Prove $aH=bH$ implies $Ha^{-1}=Hb^{-1}$

I'm trying to prove that with $H$ a subgroup of $G$ that: $$aH=bH \implies Ha^{-1}=Hb^{-1}$$ which I tried by doing the following: If $aH=bH$ means that: $ x\in aH \iff x\in bH $ then we can ...