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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2answers
26 views

Show that $\exists q\in\mathbb{Q}:d-c>|q-\sqrt{2}|$ where $d>c\in\mathbb{R}$

I'm trying to show that if $d-c>0$, then $\exists q\in\mathbb{Q}:d-c>|q-\sqrt{2}|$. In the case where $d-c>\sqrt{2}$, we have: $$ \exists q\in\mathbb{Q}:\sqrt{2}>q>0 \implies ...
-1
votes
0answers
25 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, ...
1
vote
0answers
66 views

Prove that “No one likes Reggae music” is the same as “Everyone does not like Reggae music”.

I interpreted this as a case of the extension of De Morgan's Law to quantifiers. https://en.wikipedia.org/wiki/De_Morgan%27s_laws#Extensions I know that similar questions have been asked before about ...
1
vote
0answers
13 views

Proof for two monic polynomial gcds, $d$ and $d_0$, if $d|d_0$ and $d_0|d$, then $d=d_0$

This is an extension to this, that is covered in my higher linear algebra course. I know if $d$ and $d_0$, both $\in \mathbb{F}[x]$ are monic and gcds of some polynomials $g$ and $f$ in ...
1
vote
1answer
19 views

Prove that if $\forall A \in \mathcal F (B\subseteq A)$ then $B \subseteq \bigcap \mathcal F $

This is Velleman's exercise 3.3.10. Suppose that $\mathcal F$ is a nonempty family of sets, B is a set, and $\forall A \in \mathcal F (B\subseteq A)$. Prove that $B \subseteq \bigcap \mathcal F $. My ...
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
2answers
47 views

Show by induction that $F_n \geq 2^{0.5 \cdot n}$, for $n \geq 6$

I have the following problem: Show by induction that $F_n \geq 2^{0.5 \cdot n}$, for $n \geq 6$ Where $F_n$ is the $nth$ Fibonacci number. Proof Basis $n = 6$. $F_6 = 8 \geq 2^{0.5 \cdot ...
0
votes
0answers
26 views

Singularity type for ratio of functions

Let $f,g:\mathbb{C}\to\mathbb{C}$ be two functions with a pole of order $n$ in $z_0$. I need to classify the singularity type of $\frac{f(z)}{g(z)}$ in $z_0$. I would say (intuitively) that $f \over ...
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 ...
1
vote
1answer
72 views

Are ther situations when 3 points do not lie on a circles?

Consider 3 points on a plane, points are real. Is it possible that the points are placed in a way that makes it impossible to draw a circle trough them. I know that if the point forms a line then ...
2
votes
1answer
40 views

How to prove that $f(x)x - \int_{0}^{x}{f(t) \,dt} = \int_{f(0)}^{f(x)}{f^{-1}(t) \,dt},$ for all invertible functions.

A while ago, I found that: $$f(x)x - \int_{0}^{x}{f(t) \,dt} = \int_{f(0)}^{f(x)}{f^{-1}(t) \,dt}.$$ I managed to prove it for a few functions, and I believe that it may be the case for all ...
0
votes
1answer
28 views

Analytic version of Hahn-Banach using geometric version

When studying the Hahn-Banach theorem, one can demonstrate the geometric version from scratch and use it to prove the analytic version, as is outlined in Hahn-Banach theorem: 2 versions. To do so, it ...
1
vote
2answers
44 views

Why is the topology on $\mathbb{R}$ formed by the basis $[a,b)$ normal?

Why is the topology on $\mathbb{R}$ formed by the basis $[a,b)$ normal? I need to prove that two disjoint closed sets are contained wtihin two open disjoint sets. First, I tried to understand how a ...
-1
votes
1answer
104 views

Injections, Surjections, Bijections [on hold]

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. ...
-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 ...
-4
votes
0answers
54 views

Is there a simpler alternative to LaTex for math proof writing? [closed]

I do not want to write math proofs on papers. LaTeX seems more than capable, but a person can't easily read LaTeX before it is rendered. I investigated into AsciiMath, but it generates MathML on web ...
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
117 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
44 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
72 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 ...
3
votes
4answers
99 views

$S$ and $T$ are two sets. Prove that if $|S-T|=|T-S|$, then $|S|=|T|$.

Here is the problem that I am currently working on: $S$ and $T$ are two sets. Prove that if $|S-T|=|T-S|$, then $|S|=|T|$. I have access to the answer for this proof, and wanted help with the first ...
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 ...
2
votes
3answers
50 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 ...
1
vote
0answers
24 views

How to prove partial ordering formally?

The question is: The set $S$ is defined as $\varnothing \in S$, If $x \in S$, then also $\{x\} \cup x \in S$. Prove or disprove it is partial ordering. So the set $S$ looks ...
2
votes
2answers
30 views

Why is this proof for an arbitrary function constrained to a constant one?

Sorry if this seems trivial, I'm having some difficulty understanding a proof. I'm doing exercise 5.1.14 of Velleman's How to Prove It and a solution posted in this question, including the comments, ...
1
vote
2answers
64 views

How can I be more confident that my proof is correct? (Real Analysis)

I am going through a textbook to prepare for Real Analysis and I recently tried the problem: Let $w\in\mathbb{R}$ be an irrational positive number. Set $A = \{ m+nw \mid m+nw > 0, ...
1
vote
1answer
39 views

What theorem is this? (in PDE)

I'm confused because it's titled as "Gauss's theorem about heat flux" (not in English though, I'm translating), but instead of the heat equation there's Laplace's equation written above the theorem. ...
0
votes
1answer
18 views

Closed communicating class

Let $P_{ij}$ a transition matrix, a class $C$ is closed if given two different states $i$ and $j$ $$i\in C, i\rightarrow j\Rightarrow j\in C$$ If a Markov Chain is irreducible the transition matrix ...
0
votes
2answers
59 views

Velleman's exercise $3.1.7$

Prove that if $a^3>a$ then $a^5>a$. Velleman gives this "hint": $$\text{One approach is to start by completing the following equation:}\ (a^5-a)=(a^3-a) \cdot x$$ I don't understand this ...
1
vote
1answer
44 views

Logical equivalence - Russell's Paradox

In 'How to Prove it' Velleman creates the following set: $R = \{A\in U| A \notin A \}$. This is, according to Velleman, equivalent to $\forall A \in U (A \notin A \iff A\in R) $. That is clear. ...
1
vote
2answers
59 views

Can someone explain to me why set proof involve the words “or” and “and”

For example, on proving the distributive law of set theory, the following constitutes as a proof Proof : I am new to proof involving sets but this to me seems nothing more than replacing unions ...
0
votes
4answers
38 views

Geometry question about lines

If I have two points in euclidean space or the Cartesian plane whichever and both points lie on the same side of a straight line. Both above or both below- how can I show that the segment connecting ...
4
votes
1answer
34 views

Let $\ f_1:A \rightarrow B$ and $\ f_2:A \rightarrow B$. Prove or disprove $f_1 \cap f_2$ iff $f_1=f_2$.

Here is the question I am working on (screenshot): So, I haven't worked with function proofs very much (especially in the context of iff statements and with intersections). I am looking to see ...
2
votes
3answers
31 views

Independent Poisson process

Suppose that $\{N_1(t),t\geq0\}$ and $\{N_2(t),t\geq0\}$ are independent Poisson Process with rates $\lambda_1$ and $\lambda_2$. Show that $\{N_1(t)+N_2(t),t\geq0\}$ is a Poisson process with ...
2
votes
2answers
32 views

Poisson Process proof that

For a Poisson process show, for $s<t$ that $$P(N(s)=k\mid N(t)=n)={n\choose k}\left(\frac{s}{t}\right)^k\left(1-\frac{s}{t}\right)^{n-k},\space > k=0,1,\dots,n$$ I tried a few things but ...
0
votes
1answer
21 views

Dividing by epsilon and taking the limit of epsilon to zero

In deriving the joint distribution of two order statistics, there is the following step (F(x) is the Cumulative Dist Function at x, f(x) is the PDF at x): $$ ...
2
votes
8answers
117 views

Prove by induction that $\frac{n^3}{3}+\frac{2n}{3}$ is an integer. [duplicate]

The question that I am working on is: Prove that $\dfrac{n^3}{3}+\dfrac{2n}{3} \in \mathbb Z \ \forall \ n \in \mathbb N$ The method that I think would be will work for this question is that I ...
0
votes
2answers
54 views

Induction question regarding Universe

I was given a question that looks like this. Prove that for each $2 \le n\in \mathbb N$, if $X_1,\ldots,X_n$ are subsets of some universe $U$, then the following is true: $$(X_1 \cup\cdots\cup ...
1
vote
3answers
68 views

Induction Proof using factorials

Recall that for $n \in N$, $n! = 1 \cdot 2 \cdots n$. Prove the following for each $n \in N$: $$\frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!} + \cdots + \frac{n}{(n+1)!} = 1 - \frac{1}{(n+1)!}$$ I ...
0
votes
4answers
74 views

Prove for each $n\in \mathbb{N}, 1^3 + 2^3 +\cdots+ n^3 = \frac{n^2(n+1)^2}{​4} ​ ​​​$ [duplicate]

So I was given a proof by induction question and here is my attempt $$1^3 + 2^3 + 3^3+\cdots+n^3= \frac{n^2(n+1)^2}{4}$$ $n=1$ $1=1$ Induction step: Assume statement is true for $n=k$, show true ...
6
votes
2answers
42 views

Sujection, finite set, $|X| \le n$? [closed]

Suppose that $\{1, 2, \dots, n\} \to X$ is a surjection. How do I show that $X$ is a finite set and that $|X| \le n$?
0
votes
1answer
79 views

Proof of The Basis Theorem in Linear Algebra [closed]

So I saw the following proof of the Basis Theorem in Leon Simon's "An Introduction to Multivariable Mathematics". I was wondering if anyone could help explain what is happening in it. I understand ...
0
votes
1answer
40 views

Convergent conjecture: What is the proof?

Lets say that $\def\nn{\mathbb{N}}$$\def\rr{\mathbb{R}}$$K : \nn \to \rr$ and $\displaystyle \sum_{i=1}^\infty \frac{K(i)}{K(i+1)}$ is a convergent sum. My conjecture is that the function $K$ must be ...
1
vote
1answer
39 views

Let $A,B$, and $C$ be sets. If $A\subseteq B$, $B\subseteq C$, and $C\subseteq A$, then $A=C$.

Here is my abstract maths problem. Let $A,B$, and $C$ be sets. If $A\subseteq B$, $B\subseteq C$, and $C\subseteq A$, then $A=C$. I am asked to either prove or disprove this statement. I am a little ...
2
votes
0answers
92 views

Deformation retract of a triangle

Let $X \subset \mathbb{R}^2$ be a triangle equipped with the topology induced by the euclidean topology on $\mathbb{R}^2$ and let $Y \subset X$ be the subset made of two sides of the triangle. I need ...
1
vote
1answer
96 views

(Ir)rationality of Real Numbers

I am working on this proof and this is what I got so far, can someone help me verify if what I have done is right? For all real numbers $x$ and $y$, if $x+y$ is rational and $x-y$ is irrational ...
0
votes
1answer
46 views

Constructive Induction to derive and prove the formula for a geometric sequence

$\sum_{i=1}^nr^i = a \cdot (b^n) + c$ Base Case: n=1, this holds Inductive Hypothesis: Assume for $n = k$, $k \ge 1$ that $\sum_{i=1}^k r^i = a * (b^k) + c$. Inductive Step: Prove for $n = k+1$ ...
1
vote
2answers
23 views

Proof: Splitting triangle

Prove that, if ABC is a triangle with Angle A= 90 Degrees, Angle B=30 Degrees and C= 60 Degress, and W is the midpoint of the hypotenus, then the line connetcting W to A divides ABC into an ...
3
votes
1answer
32 views

Doubly stochastic matrix proof

A transition matrix $P$ is said to be doubly stochastic if the sum over each column equals one, that is $\sum_i P_{ij}=1\space\forall i$. If such a chain is irreducible and aperiodic and ...