For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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2
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4answers
33 views

Use class algebra to prove the following: If A∩B = ∅ and A∪B = C, then A = C-B

I'm having a bit of trouble proving the following. If A∩B = ∅ and A∪B = C, then A = C-B My initial attempt is to prove it directly, however, I believe I'm assuming the consequent, namely, A = C-B, ...
3
votes
2answers
73 views

Is it possible to find such a $f$?

I search a continuous function $f : [0,+\infty[ \to \mathbb{R}$ such as : $\lim \limits_{x\to +\infty} \frac{1}{x} \int \limits_{0}^{x} f(t)\mathrm{d}t=\pm \infty$ and ($\lim \limits_{x\to ...
3
votes
3answers
62 views

Prove that if $\mathcal{F} \subseteq \mathcal{G}$ then $\cup\mathcal{F} \subseteq \cup\mathcal{G}$

Suppose $\mathcal{F}$ and $\mathcal{G}$ are families of sets. Prove that if $\mathcal{F} \subseteq \mathcal{G}$ then $\cup\mathcal{F} \subseteq \cup\mathcal{G}$ My attempt: Given $\mathcal{F} ...
1
vote
2answers
64 views

proof of number of prime factors of $n$

Given an integer $n$ between 1 and 1000000, how do you directly prove that $n$ has at most 19 prime factors (with multiplicity)? I'm quite stuck on how to do this. I can understand the base case ...
1
vote
1answer
30 views

Extending a Field Monomorphism

In theorem A3.5 of Ash's book Abstract Algebra: The Basic Graduate Year (page 20 in this pdf), the author set out to prove the following. Let $\sigma: F \rightarrow L$ be a field monomorphism ...
3
votes
4answers
71 views

Prove by induction: $\frac{1}{2!}+\frac{2}{3!}+\cdots+\frac{n}{(n+1)!}=\frac{n!-1}{n!}$

Prove $$\frac{1}{2!}+\frac{2}{3!}+\cdots+\frac{n}{(n+1)!}=\frac{n!-1}{n!}.$$ My problem with this is that it doesn't hold for the base case: $n=1$. This question is from the book "Abstract ...
3
votes
5answers
56 views

Trigonometry identity $\csc x\cot x=\frac{\cos ^3x}{\sin^2 x}+\cos x$

How to prove that $\csc x\cot x=\frac{\cos ^3x}{\sin^2 x}+\cos x$? I tried manupulating the left hand side but ended up in $\frac{\cos x}{\sin^2 x}$. Can someone show me? Thanks in advance.
0
votes
3answers
84 views

Please Help me understand this proof

DOUBT What i didnot understand is from where it is written our new goal means there exists a... I didnot understand how there exists word popped up here and why the new givens are written as they ...
-1
votes
0answers
26 views

Could someone give a detailed (yet elementary) proof for Jensen's inequality?

I want to prove that Suppose there is a function $f:[a,b] \to \mathbb R$, and there are $x_i \in [a,b], w_i \gt 0 $ for $i=1,\dots,n$ such that $\sum_{i=1}^nw_i=1$, then if the function is convex, ...
2
votes
5answers
52 views

Showing that $\frac{1}{2^n +1} + \frac{1}{2^n +2} + \cdots + \frac{1}{2^{n+1}}\geq \frac{1}{2}$ for all $n\geq 1$

Show that $$\frac{1}{2^n +1} + \frac{1}{2^n +2} + \cdots + \frac{1}{2^{n+1}}\geq \frac{1}{2}$$ for all $n\geq 1$ I need this in order to complete my proof that $1 + \frac{n}{2} \leq H_{2^n}$, but ...
0
votes
3answers
24 views

Prove that if $U$ is an orthogonal $n\times n$ matrix, then the rows of $U$ form an orthonormal basis for $\mathbb{R}^n$

Prove that if $U$ is an orthogonal $n\times n$ matrix, then the rows of $U$ form an orthonormal basis for $\mathbb{R}^n$ I'm unsure how to proceed with proving this. Basically my idea is as ...
0
votes
1answer
51 views

Is that form of Cesàro's theorem correct?

First, for sequences, we know that : "If a sequence $(a_n)_{n \ge 1}$ converges to $l\in \mathbb{R}$, then the sequence $(b_n)_{n \ge 1}$ defined by : $b_n=\frac{1}{n} \sum \limits_{k=1}^{n}a_k$ ...
2
votes
0answers
25 views

Proofs by analysing games

I recently read the following article giving a novel proof of the uncountability of $\mathbb{R}$ by analysing a particular game, amongst other results. ...
6
votes
4answers
56 views

Proving $\sum_{i=1}^n 2^i = 2^{n+1} - 2$ using strong induction [duplicate]

I just started learning proof by induction in class, but got a problem requiring proof by strong induction. Here is the problem. Prove by strong induction: $$\sum_{i=1}^n 2^i = 2^{n+1} - 2$$ ...
1
vote
3answers
78 views

How to show that a statement in sets is false?

How to show that a statement in sets is false and prove its negation is true? For example I have the exercise: Let's say that $E$ is a non-empty set and $A,B,C$ $\subseteq$$E$.For each $Α,Β,C$ how ...
5
votes
2answers
63 views

Proving that $(A\setminus C)\cap(B\setminus C)\cap(A\setminus B)=\emptyset$

For each $A,B,C$ how would I prove that $(A\setminus C)\cap(B\setminus C)\cap(A\setminus B)=\emptyset$ ? My thoughts are if $x\in (A\setminus C)\cap(B\setminus C)\cap(A\setminus B)$, then $x\in ...
1
vote
1answer
87 views

Why prove that area is unique?

in the book Apostol's Calculus Volume 1, in the proof of the area of under of the parabola $x^2$ from $x=0$ to $x=b$ it is shown that the area $A$ must satisfy ...
3
votes
0answers
62 views

Is my proof rigorous? (Archimedes area of parabola)

I am currently reading Apostol's Calculus volume 1 and was revising the part where the area of a parabolic segment is found. I decided to write my own proof similar to the one in the book, which I ...
3
votes
0answers
57 views

List of techniques to evaluate limits?

I'd like to make a complete list of techniques to solve a limit. Definition of the limit Continuous functions Algebra of limits Addition, multiplication, division Composition Inverse function ...
1
vote
1answer
53 views

A consequence of Cesàro's theorem

Here is the statement : "Let $(a_n)_{n\ge 1}$ a real or complex sequence and $l \in \bar{\mathbb{R}}$. If $\lim \limits_{n\to +\infty} a_{n+1} - a_{n}=l$, then $\lim \limits_{n\to ...
0
votes
1answer
27 views

Are there infinitely many $k$ for which $\frac{\sigma(k)}{k}$ is a rational square where $ \sigma(k) $ and $k$ both are square?

Is There some one who can show me if there are infinitely many $k$ for which $$\frac{\sigma(k)}{k}$$ is a rational square where $\sigma(k)$ and $k$ both are square ? Note :$\sigma(k)$ is sum ...
0
votes
1answer
26 views

Proof that a real number must occur in the domain between two other real numbers

I saw the question stating that if your speed is $v_0 = 0$ km/h at $t=0$ and your speed is $v_{30} = 20$ km/h at $t=30$, then did you ever had a speed of $v=\pi$? Obviously this is the case as speeds ...
1
vote
1answer
30 views

How do I show that this topology on this linearly-ordered set is regular?

Given some linear ordered set $X$, we define a topology by the basis: all sets of the form $(a,b)$ or $(a,\infty)$ or $(-\infty,b)$, where $a,b \in X$. I need to prove that this topology is regular, ...
0
votes
0answers
39 views

Help fix this proof.

What is wrong with this proof? I followed the example of the answer to another one of my questions, here Define a general recurrence relation as $$f(x)^2=A(x)+B(x)f(x+n).$$ Substitute the root ...
1
vote
2answers
48 views

Showing that the set of $2 \times 2$ real orthogonal matrices has a particular parameterization

Theorem Every orthogonal matrix in $\mathbb{R}^{2, 2}$ is in the form \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix} or \begin{bmatrix} \cos\theta ...
1
vote
5answers
58 views

Proving that that ${(R \setminus S)\setminus T} \subseteq R \setminus (S \setminus T)$

How might I prove that ${(R \setminus S)\setminus T} \subseteq R \setminus (S \setminus T)$? I am not sure the best place to start other than assuming $x\in(R \setminus S)\setminus T$ and trying to ...
1
vote
3answers
27 views

Finding all the divisors of $a$ by decomposing it into the product $p^{\alpha_1}_{1} \cdot p^{\alpha_2}_{2} \cdots p^{\alpha_r}_{r}$

I'm trying to prove the following statement regarding the fundamental facts of prime numbers, but I don't really understand the relationship between $a$ and $b$. In order to find all the divisors ...
6
votes
1answer
341 views

Application of Jensen's inequality to $x^x+y^y+z^z$

Claim: If $x, y, z >0$ and $x+y+z = 3\pi, $ then $x^x + y^y + z^z > 81.$ My attempt: Let $f(w) = w^w$, so $f$ is convex on $(0, \infty).$ By Jensen's inequality, $f(x\frac{x}{3\pi}+ ...
0
votes
1answer
76 views

Find the number of flags of different types using induction

A flagpole is $n$ feet tall. On this pole we display flags of the following types: red flags that are $1$ foot tall, blue flags that are $2$ feet tall, and green flags that are $2$ feet ...
-1
votes
1answer
106 views

Injections, Surjections, Bijections [closed]

So i was given a question that asks me to determine whether the function is injective, bijective, or surjective. If you answer bijective than determine the functions inverse, domain, and target space. ...
0
votes
1answer
49 views

A vessel contains $x$ amount of milk out of which $y$ amount is taken out and replaced with water $n$ times.

There is a formula in my book for questions of type, A vessel contains $x$ amount of milk out of which $y$ amount is taken out and replaced with water. After $n$ such operations what will be the ...
0
votes
0answers
33 views

Demonstrative geometry around the world and its significance.

This is not exactly a mathematical question. I am from Pakistan; and over here students are taught a subject 'demonstrative geometry' (as a part of mathematics) from secondary level education. ...
2
votes
4answers
58 views

Prove by induction that $\sum_{k=1}^{n} k^3 = \bigg( \sum_{k=1}^{n}k\bigg)^2$ [duplicate]

Show the following for all positive integers using proof by induction: $$\sum_{k=1}^{n} k^3 = \bigg( \sum_{k=1}^{n}k\bigg)^2$$ Base case (n = 1) passes: $1^3 = 1^2$ We assume the following: ...
3
votes
1answer
49 views

Methods to Minimize Functions and Integrals over $\mathbb{N}$.

In a paper I'm writing, I have to minimize a messy function $f(\mu,n)$ where $\mu \in \mathbb{R}$ and $n \in \mathbb{N}$. That is, given $\mu \in \mathbb{R}$, I need to minimize the one variable ...
3
votes
4answers
101 views

Show that the standard integral: $\int_{0}^{\infty} x^4\mathrm{e}^{-\alpha x^2}\mathrm dx =\frac{3}{8}{(\frac{\pi}{\alpha^5})}^\frac{1}{2}$ [duplicate]

In my physics course this standard formula is used a lot without proof so it would be interesting to see a neat proof for it. From a previous thread by me I know the proof for $\int ...
2
votes
1answer
39 views

Proof for $\mathbf{M}$ unitary if $||\mathbf{M}\mathbf{v}|| = ||\mathbf{v}||$

Let $\mathbf{M} \in \mathbb{F}^{n, n}$. Then $||\mathbf{M}\mathbf{v}|| = ||\mathbf{v}||$($\mathbf{v} \in \mathbb{F}^n$) implies that $\mathbf{M}$ is unitary. My question is, how to prove this ...
0
votes
2answers
42 views

Use proof by contradiction to prove that if $a$ and $b$ are odd integers, then $4 \nmid (a^2+b^2)$.

I have been working through the following proof: Use proof by contradiction to prove that if $a$ and $b$ are odd integers, then $4 \nmid (a^2+b^2)$. Below, I have included screenshots of the ...
-1
votes
0answers
70 views

Determine whether it is injective, surjective, bijective or neither injective nor surjective [closed]

The question i was given asked (a) Determine whether it is injective, surjective, bijective or neither injective nor surjective. (b) If you answered "bijective" in part (a) determine the ...
0
votes
1answer
25 views

defining a sequence of numbers L n≥1, and prove something about it

So i was given this question just to try for practice to test our knowledge and i'm really confused on how to go about this question. How should i go about this question, i'm confused with the greek ...
3
votes
4answers
120 views

Proving that if $a+b$ and $ab$ are of the same parity, then $a$ and $b$ are even.

Here is the proof: Let $a,b \in \mathbb{Z}$. Prove that if $a+b$ and $ab$ are of the same parity, then $a$ and $b$ are even. When working these problems, I do try to set them up logically. My ...
-4
votes
0answers
46 views

prove that for any 2n≥2 and any \a ​1 ​​ ,…,a ​n ​​ ∈N, we have the following: [closed]

So the question I was given goes like this we will introduce a mystery function,P:N→N. We don't know a formula for P (and we won't be able to determine one!) but we do know that P satisfies the ...
1
vote
3answers
73 views

Suppose $A\subseteq \mathscr P (A)$. Prove that $ \mathscr P (A)\subseteq \mathscr P ( \mathscr P (A))$

This is Velleman's exercise 3.3.4. Suppose $A\subseteq \mathscr P (A)$. Prove that $ \mathscr P (A)\subseteq \mathscr P ( \mathscr P (A))$. I started reexpressing the terms in their equivalent forms ...
2
votes
3answers
74 views

Proving $10\cdot n=0$ for all $n\in\mathbb{Z}$ with $n\geq 0$ using strong induction

The question says $10\cdot n=0$ for all $n\in\mathbb{Z}$ with $n\geq 0$. Here is my proof by strong induction: Base case: $10\cdot0=0$. Let $k\geq 0$, and suppose that for any $m\leq k$ we have ...
0
votes
1answer
43 views

Estimate for the integral using convexity bound

I'm reading the proof of Hardy and Littlewood's theorem in the book Analytic Number Theory, written by Henryk Iwaniec and Emmanuel Kowalski (p. 547): Theorem (Hardy and Littlewood): Let $N_0(T)$ ...
1
vote
1answer
46 views

Prove or disprove: $(\mathbb{Z}^*, \cdot)$ and/or $(\mathbb{Z}^*, \div)$ is a group.

I am teaching myself information about groups, but don't really understand how to work through this problem. Here is what I have been thinking so far (please note that I do not need to work through ...
1
vote
1answer
141 views

Interesting property of Pascal's Triangle

I was looking at the Pascal's Triangle and saw that for all central numbers in even length row $a \gt 17$, the number $\dbinom{a}{b-2}$ is greater than $\dbinom{a-1}{b}$. This is where $b$ is equal to ...
0
votes
1answer
30 views

Interesting Combinatorics question relating the coefficients of variables in Pascal's Triangle

I tried this problem for a while by canceling the factorials on either side but for whatever reason, wasn't able to solve it. Could someone please help me? Is there a proof that ...
2
votes
2answers
24 views

creative method to obtain range of newton function ?!

I am searching for more proof that the range of $y=\frac{x}{x^2+1}$ is $ \frac{-1}{2}\leq y \leq \frac{+1}{2}$ these are my tries : domain is $\mathbb{R}$ first : $$y=\frac{x}{x^2+1}\\yx^2+y=x ...
0
votes
0answers
31 views

Proof Check In: Prove that $(\mathbb{Z}_n, +)$, the integers (mod $n$) under addition, is a group.

I received some help and direction on this from some users a few days ago, and have tried to take that information and craft it into something proofy. I would appreciate general suggestions, edits, ...
2
votes
3answers
51 views

Prove that if $x \neq 0$, then if $ y = \frac{3x^2+2y}{x^2+2}$ then $y=3$

The problem is the following (Velleman's exercise 3.2.10): Suppose that $x$ and $y$ are real numbers. Prove that if $x \neq 0$, then if $ y = \frac{3x^2+2y}{x^2+2}$ then $y=3$. My approach so ...