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

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1answer
29 views

Understanding a proof about nested nonempty connected compact subsets

I know this question has asked to death here on MSE but I have not found a satisfactory solution. A solution found online is extremely elegant but I do not quite understand it! Given nested ...
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2answers
28 views

Proving two integers of opposite parity have an even product?

I think I might be beginning to wrap my head around some simpler proofs, but I'm a little stumped on this one from my textbook: Use a direct proof to show that if two integers have opposite ...
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1answer
42 views

For all sets A and B, if A ⊆ B, then A ∪ B ⊆ B

I am trying to solve a proof, but I'm a little lost on how to structure it. I have the following setup, but I'm not sure what to put in most of the blank spaces. Proposition: For all sets A and B, ...
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4answers
37 views

prove that $-1 \le \frac{ac+bd}{\sqrt{a^2+b^2}\sqrt{c^2+d^2}} \le 1$

For the real numbers $a, b, c, d$ prove that $$-1 \le \dfrac{ac+bd}{\sqrt{a^2+b^2}\sqrt{c^2+d^2}} \le 1$$ Actually if we let $\vec{u} = (a, b)$ and $\vec{v} = (c, d)$ then by dot product we got ...
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0answers
12 views

Sufficient condition for upper semicontinuous functions

My question might be fundamental but I'm glad if you give some help since I don't find any idea. Let $X$ be a bounded and closed subset of $\mathbb{R}$. A function $f:X\to\mathbb{R}$ is called to be ...
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2answers
38 views

Proving the Trichotomy Property

I need to show that if $a,b\in \mathbb{R}$, then only one of the following holds: $a\in \mathbb{P}, -a\in \mathbb{P}$, or $a=0.$ By a definition in my book, if $a-b \in \mathbb{P}$, then $a>b$; ...
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0answers
12 views

Upper estimate between an original function and its sup-convolution under a limitation

My setting maybe look rather special but I'm glad if you give some answers. Let $f:[0,1]\to\mathbb{R}$ be a bounded, upper semicontinuous function and $f^{\varepsilon}:[0,1]\to\mathbb{R}$ be $f$'s ...
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0answers
46 views

Is it correct? $1^n +2^n +…+(p-1)^n=-1 \pmod p$

$p$ a prime number, $n\in \mathbb{N} $ and $p-1\mid n$ then $1^n +2^n +...+(p-1)^n=-1 \pmod p$ I'm not sure if my proof is correct: Take the group $G=(\mathbb{Z*}_{p},\cdot)$ with the ...
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2answers
37 views

Simple Proof for Commutative Property of Multiplication

I'm supposed to show that $a\cdot b=b\cdot a$ for a set $K:=\{s+t\sqrt2:s,t\in\mathbb{Q}\}$ to show that this set is a field. I was going to set it up like: Let $a, b\in K$ such that ...
2
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1answer
33 views

Permutations + Combinations Proof

$W_n^{(k)}$ is the number of permutations in a set of all $n!$ permutations in a $n$-element set which has $k$ fixed points. $W_n^{(0)}$ is the number of n-derangements where $\frac{W_n}{n!} ...
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2answers
26 views

Forming a conjecture about the relationship between $g(A \cup B)$ and $ g(A) \cup g(B)$ for an arbitrary function g.

Question: Form and prove a conjecture about the relationship between $g(A \cup B)$ and $ g(A) \cup g(B)$ for an arbitrary function g. I've already shown this: For an arbitrary function $g$: ...
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2answers
22 views

Proof using pumping lemma that $\{0^m1^n \mid m \neq n \}$ is not regular

First of all, there's already a question very similar to this, but in my case I just wanted to show my attempt with the hope that if there are any erros or it's wrong completely you can help me in the ...
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4answers
54 views

Prove that for every number that's not a prime, it exists a prime $p$ with $p\mid n$ and $p \leq \sqrt{n}$

For $n \in \mathbb{N}$, if $n$ is not a prime and $n ≥ 2$, it exists a prime $p$ with $p\mid n$ and $p \leq \sqrt{n}$. How would I mathematically correctly prove this sentence? I've thought about ...
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2answers
38 views

Proving a decreasing, convergent sequence.

Let $t_1=4$ and $t_{n+1}= \sqrt{3+2t_n}$. How do I prove that $3<t_{n+1}<t_n$? I understand that it is true, but I don't understand how to show it... Thanks!
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0answers
9 views

Show that {$x \in I | f(x) = g(x)$} is closed using the Global Continuity Theorem

Let $I = [a, b]$ and let $f, g : I \to R$ be continuous functions on I. Show that {$x \in I | f(x) = g(x)$} is closed using the Global Continuity Theorem. My classmate and I have been struggling with ...
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0answers
37 views

Need proof assistance for theorem involving GCD theorem

I've tried this proof every which way I can think of and I can't figure out how to word it; am I on the right track? Problem: Let $f, g, s, m$ be integers. If $gcd(f, g) = s$ and both $f$ and $g$ ...
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1answer
33 views

Condition under which a point belongs to the convex hull of some other points

I can't understand a fact used in the proof of a theorem I am reading. It can be stated as follows: Let $x_0,\dots,x_n$ be points in some Euclidean space $\Bbb R^d$. Then a point $p\in\Bbb R^d$ ...
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1answer
53 views

Showing there are no nontrivial ring homomorphisms from $\mathbb{Z}\to\mathbb{Z}$ [duplicate]

I have: If $\phi:\mathbb{Z}\to\mathbb{Z}$ is a homomorphism, then $f(1)=f(1\ast1)=f(1)\cdot f(1)$. Then $$0=f(1)\cdot f(1)-f(1)=f(1)\cdot\left[f(1)-\epsilon\right]$$ implies that $f(1)=0$ or ...
4
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1answer
44 views

$\int_{\Omega} |f_n-f||f_n| \, d \mu \to 0$ if $f_n \in L^1(\Omega)$, $f_n \to f$.

Suppose $f_n \to f$ in $L^1(\Omega)$ where $\mu(\Omega)=1$. Suppose $$\int_{\Omega} |f_n| \, d\mu \leq M$$ for all $n$. Is there a way to show that the integral $$\int_{\Omega} |f_n-f||f_n| \, d ...
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0answers
19 views

If $E \subset [0,1]$ satisfies, for any $I \subset [0,1]$, $m(E \cap I) \geq \frac{1}{2}m(I)$, then $m(E)=1$

If $E \subset [0,1]$ satisfies, for any $I \subset [0,1]$, $m(E \cap I) \geq \frac{1}{2}m(I)$, then $m(E)=1.$ I'm aware this post exists elsewhere, say, here but what I don't understand is why we ...
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0answers
26 views

Let $n\in \mathbb{N}$ and $p>2$ a prime number show that $(1+p)^{p^n} \equiv 1+p^{n+1} \ [p^{n+2}]$

I tried an induction on $n$ : For $n=0$, we obtain : $1+p \equiv 1+p \ [p^2]$ it is right ! For $n=1$, I get : $(1+p)^p = \sum \limits_{k=0}^p \binom pk p^k$ and I noticed that for $k\in ...
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2answers
19 views

Given a forest, adding k edges would result in a cycle Proof

Assume you have a forest with k connected components. Prove that if you added $k$ edges, you would obtain a cycle. I’m thinking these facts/theorems may be useful... In a forest, each component ...
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0answers
29 views

How to prove members of this series differ from an integer by, at most, 1/n?

Consider the series , where a is a positive real number. a, 2a, 3a, .... (n-1)a Prove that there is one member of this series that differs from an integer by at most 1/n. My approach : Draw a ...
0
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2answers
44 views

Prove or disprove: $g(x) = x^2 - 2x + 1$ monotonically increases for $x > 1$.

I know I can compare $g(x)$ and $g(x+a)$ where $x$ is in the region of interest and $a > 0$, and to expand out the algebra to show that $g(x+a)$ always equals or exceeds $g(x)$ but I'm not ...
0
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1answer
71 views

Proof for the length of plane curves

Given that the plane curve is a parametric curve with equations $x=f(t)$ and $y=g(t)$ and we are finding the length of the curve in the interval (a,b) and so we divide the given plane curve into ...
1
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1answer
27 views

Multiplication Principle Proof

I am trying to prove the following; If $X$ and $Y$ are finite, then $|X \times Y| = |X||Y|$. Now, I'll define a bijection $g:\mathbb{N_{n}} \rightarrow X$ and a bijection $f: \mathbb{N_{m}} ...
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2answers
73 views

Prove that $3(\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b}) \ge 10 + 8\cdot \frac{a^2+b^2+c^2}{ab+bc+ca}$

For the positive real numbers $a, b, c$ prove that $$3\bigg(\dfrac{a+b}{c}+\dfrac{b+c}{a}+\dfrac{c+a}{b}\bigg) \ge 10 + 8\cdot \dfrac{a^2+b^2+c^2}{ab+bc+ca}$$ I did the following: ...
0
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2answers
27 views

Adjacency Matrix of a Graph Length of Paths Proof

Let A be an adjacency matrix of a graph G. Prove that the (i, j)th entry of $A^2$ is the number of paths of length 2 between vertex i and vertex j. *I know the adjacency matrix will be a square ...
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3answers
42 views

$\operatorname{gcd} \, (a,b) = 1$ then $\operatorname{gcd} \, (a^n,b^k) = 1$

Statement: Suppose $(a,b) = 1$ then $(a^n,b^k) = 1$ for $n,k \geq 1$. Attempt at Proof: Let $P$ be the set of all primes. Let $P_a$ be the set of primes $p_i$ such that $$a = \prod_{i=1}^{r_1} ...
0
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1answer
42 views

Use calculus to show the derivative of the area of the circle is the circumference of the circle

Let $A(r)$ denote the area of the circle of the radius $r>0$, and let $C(r)$ denote the circumference of the circle, show $A'(r)=C(r)$ for all $r>0$. I found a similar question which has ...
1
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1answer
18 views

If $f(n) \geq g(n)$ for all $n \geq n_0$ then $f(n) \geq cg(n)$ for all $n \geq n_1$

So I've thought of this statement which I'd like to know if it's true but I can't seem to prove it: If there exists a number $n_0$ such that $$f(n) \geq g(n)$$ for all $n \geq n_0$ then there is a ...
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2answers
65 views

Let a,b be rationals and x irrational. Show that if $\frac{x+a}{x+b}$ is rational, then $a=b$.

I'm trying to solve the following problems: Let $a$,$b$ be rationals and $x$ irrational. Show that if $\frac{x+a}{x+b}$ is rational, then $a=b$ Let $x$,$y$ be rationals such that ...
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0answers
37 views

Proving the Commutativity of Set intersection.

Hi this is basically the question: Write down a formula which states that for any two sets $X$ and $Y$ , the set $X \cap Y$ is the same as the set $Y \cap X$. Then, prove this statement. The union ...
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2answers
33 views

Proving additive inverse of vector set exists and “works”

Let V = {$a_1, a_2): a_1, a_2 \in F$} where F is a field. Define addition of elements of V coordinate wise, and for $c \in F$ and $(a_1, a_2 \in V$}, define $c(a_1, a_2) = (a_1, 0)$. In my proof, I ...
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2answers
44 views

With 8 bits is it possible to obtain an integer in more than one way? [duplicate]

This is just a curiosity that just came to my mind while thinking at IP addresses. A byte is composed of 8 bits. A bit can either be $0$ or $1$. IPv4 addresses are composed of a group of 4 bytes. ...
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2answers
25 views

Prove that $3$ divides $n^2 + n$ iff $n$ mod $3$ $\neq 1$

I'm trying to prove that $3$ divides $n^2 + n$ iff $n$ mod $3$ $\neq 1$ . I already tried it with proving a double implication, but I did not succeed. A tip or kickstart would be great. Thank you
1
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1answer
21 views

Conditions required for Inequality to hold

Given that $0 \leq X \leq x \leq y \leq Y < \infty$, I am interested to know the condition whereby $\frac{\sqrt{y}-\sqrt{x}}{\sqrt{y}+\sqrt{x}} \leq \frac{\sqrt{Y}-\sqrt{X}}{\sqrt{Y}+\sqrt{X}} $ ...
0
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5answers
61 views

Prove that if $ax^2 + bx + c = 0$ for all values of $x$, then $a=b=c=0$

Could anyone help me out with writing this proof for my linear algebra class? My classmate told me to just plug in 3 different values for $x$ and that should prove that $a=b=c=0$. However, I could ...
6
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5answers
118 views

What is the most used method for proving continuity for simple functions such as $f(x) = x^{1/3}$

In analysis we talked about a very general definition of continuity: $f:A \to B$ is continuous if $U \subset B$ is open, $f^{-1}(U) = V \subset A $ is open Quite elegant Another definition is if ...
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3answers
47 views

Prove matrix $X'XA$ equals $X'XB$ iff $XA = XB$

I am working on a proof that $X'XA = X'XB$ if and only if $XA = XB.$ How do I finish this, and then use to prove the result of X (generalized inverse)X = X? $ X'XA = X'XB $ $ X'XA - X'XB = 0 $ ...
1
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5answers
53 views

There are infinitely many vectors such that $\|\mathbf u-(1,1,1)\|\le 3$ and $\|\mathbf u+(1,1,1)\|\le 3$

I want to prove that there are infinitely many solutions in $3$-space for $\|\mathbf u-(1,1,1)\|\le 3$ and $\|\mathbf u+(1,1,1)\|\le 3$ (where bold refers to vectors). My proof: What is wrong with ...
0
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1answer
27 views

In the following equation $Av = \lambda v$ $v$ cannot be $\vec{0}$

In this question and answers to such question people say that $v$ cannot be $\vec{0}$, but this not correct in my opinion. We assume it is different from $\vec{0}$ because otherwise $Av = \lambda v$ ...
0
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0answers
35 views

Show that absolute value satisfies triangle inequality, how? [duplicate]

I wish to show that given $a,b,c \in \mathbb{R}$, the following holds: $|a - c| \leq |a-b| + |b-c|$ Using the definition $| x | = \max\{x, -x\}$ I can't seem to be able to show that $|a - c| \leq ...
1
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1answer
28 views

How to prove that bijective functions have surjective inverses

I am trying to prove Since $f$ is bijective therefore it has an inverse and $f^{-1}$ is bijective But stuck on proving that $f^{-1}$ is surjective Suppose $f: A \to B$ is a bijection, then a ...
2
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1answer
27 views

Simple Graph Degree Sequence Proof

Let G be a simple graph with degree sequence $(d_1,d_2,...,d_n)$. Prove that for each k, $0<k<n$: $$\sum_{i=1}^k d_i\le k(k-1)+\sum_{i=k+1}^n min(k,d_i)$$ I'm new to graph theory and proof ...
2
votes
2answers
36 views

Can I use “$\iff$” symbol when I “transform” an expression to another form?

I am writing a solution to prove that $\sqrt5$ is not rational. Here is my first half proof: Assume $\sqrt{5}$ is a rational number. By the definition of rational number, $\sqrt{5} = \frac{p}{q}$, ...
0
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0answers
66 views

Algorithms in formal logic + ZFC

From my understanding, most of mathematics can be built up assuming a mechanical procedure of manipulating finite strings of symbols according to certain rules. A conventional way to do it is via the ...
0
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1answer
33 views

Simple graph proving cardinality of a set is even

Let G be a simple graph, and let S be the set of vertices of even degree, and T be the set of vertices of odd degree. Prove that the cardinality (number of things in) T is even. I am new to graph ...
2
votes
2answers
36 views

Proof that if $S$ is closed then $S^c$ is open from Pugh's text

Recall $S$ is closed if it contains all the limit points and $S$ is open if the metric $d(p,q) < r$ around every $q \in S$ for some $r>0$ Pugh's real analysis provides a proof as follows: ...
0
votes
1answer
21 views

What is the biggest computation one should use for an induction base

First of all sorry for me (informal) language here - i am not a native english speaker. I have tried to look up my question here but couldnt find anything. I also tried to look it up on google but i ...