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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2answers
60 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
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1answer
50 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
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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 ...
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4answers
40 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
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1answer
35 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
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3answers
32 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
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1answer
23 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
119 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 ...
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2answers
56 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 ...
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3answers
74 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 ...
1
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1answer
86 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
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1answer
41 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
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1answer
40 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
125 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
100 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
52 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
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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
35 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 ...
1
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0answers
36 views

How to formally express a negative statement (in the wording or formulation of a theorem, for instance)

This is a doubt about English mathematical formal language. I would like to know the best way to express a negative hypothesis, in the formulation or statement of a theorem, proposition, etc., using ...
2
votes
3answers
71 views

How to prove $A=(A\setminus B)\cup (A\cap B)$ [duplicate]

How to prove $A=(A\setminus B)\cup (A\cap B)$. I have seen this problem and the solution is clear to me. Initially I was satisfied by my prove but now I think it is wrong. How I have proved ...
1
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2answers
65 views

Try to prove a generalization of the IVT?

Here is the statement : "Let $f: (a,b) \to \mathbb{R}$ a continuous function on $(a,b)$ with $a<b$ and $a,b \in \bar{\mathbb{R}}$. Then for all $u \in \left(\lim \limits_{x\to a} f(x), \lim ...
0
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1answer
33 views

If $ax + by = 8$, what is $\operatorname{gcd}(a, b)$?

Our instructor has given us this problem: If $ax + by = 8$, what is $\operatorname{gcd}(a, b)$? I'm confused. Is it not just $8$? Since, say, $\operatorname{gcd}(a, b) = n$, so there must exist ...
2
votes
1answer
32 views

If $P^r$ has all positive entries, then so does $P^n$

Let $P$ be the transition probability matrix of a Markov Chain. Argue that it for some positive integer r, $P^r$ has all positive entries, then so does $P^n$, for all integers $n\geq r$ I ...
0
votes
0answers
51 views

Proving transcendental numbers

I'll warn now that this is probably a big question, but I am wondering if anyone can explain why it is so difficult to prove whether a number is transcendental or algebraic. For example, it is now ...
2
votes
1answer
70 views

When is it mathematically correct to take a limit in certain expresions?

So now I managed to put together a couple of proofs, that each of them use a similar procedure in a crucial step, and I am not sure what are the requirements for this step to be true. First example: ...
1
vote
1answer
27 views

negation of a null sequence

I have that a sequence $\{a_n\}$ is null $\Leftrightarrow \forall \epsilon >0, \exists X$ such that $$|a_n| < \epsilon \ \forall n > X.$$ I want to give a definition when a sequence is not ...
2
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2answers
36 views

Proofs: Induction on Handsakes

Here is the problem: Suppose $n$ people are at a party, and some number of them shake hands. At the end of the party, each guest $G_i$, $1 \leq i \leq n$ shares that they shook hands $x_i$ times. ...
1
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2answers
45 views

An exercise problem in Axler Linear Algebra Done Right for linear mapping

So I have written down a proof but feel uncertain about whether it is valid or not, could somebody please Check it and, if there exists any gap, inform me? The question is the following (Chapter 3 ...
-1
votes
2answers
39 views

Set Theory Proof $A=B$ [closed]

Let $A$ be the set of all integers $x$ such that $x = 2k$ for some integer $k$ Let $B$ be the set of all integers $x$ such that $x = 2k+2$ for some integer $k$ Give a formal proof that $A = B$.
1
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2answers
43 views

Proof the statement

Given a finite aperiodic irreducible Markov Chain, prove that for some $n$ all terms of $P^n$ are positive. I'm little lost in how to prove that, but I know that: $i)$ If a Markov Chain is ...
1
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1answer
14 views

Confused about a quality of Existential Generalization and Instantiation

Let me preface the question with a "proof" $1. \exists yP(y) \quad Premise \\ 2. P(B) \quad \quad 1,E.I. \\ 3. \exists xP(x) \quad 2,E.G. $ However, I am not sure if it is to "safe" to say that ...
2
votes
1answer
18 views

Stochastic matrix proof

Every stochastic $n\times n$ matrix corresponds to a Markov chain for which it is the one-step transition matrix. However, not every stochastic matrix $n\times n$ is the two-step transition ...
2
votes
2answers
133 views

When does $i^x=x$

Can someone please help me solve $i^x=x$? So far I have: $$i^x=x$$ $$\frac{\ln(x)}{\ln(i)}=x$$ $$e^{i\pi}=-1$$ $$e^{i\pi/2}=i$$ $$\frac{\ln(x)}{\frac{i\pi}{2}}=x$$ $$\ln(x)=\frac{i x \pi}{2}$$ ...
0
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2answers
52 views

Prove there exists a bijective function $\left\{a,\cdots,b\right\}\rightarrow\left\{a+k,\cdots,b+k\right\}$ for $k\in\mathbb{N}$

I must prove that there exists a bijective function $\left\{a,\cdots,b\right\}\rightarrow\left\{a+k,\cdots,b+k\right\}$ for $k\in\mathbb{N}$ (this is not homework). This is the proof I've come up with ...
0
votes
1answer
23 views

Proving that a Binary Tree of $n$ nodes has a height of at least $\log(n)$.

For a homework assignment, I need to prove that a Binary Tree of $n$ nodes has a height of at least $log(k)$. I started out by testing some trees that were filled at every layer, and checking $log(n)$ ...
2
votes
2answers
30 views

Greatest common divisors equal?

Let $a,b$ be natural numbers. Show that $gcd(a^n,b^n)$ = ($gcd(a,b)^n)$ for any integer $n$. How I started was first proof by contradiction, and then tried to do an inductive proof when that didn't ...
2
votes
1answer
37 views

Parabolic range conditions proof

This problem is getting the better of me, since I have no idea where to start: The equation of a curve is $y=ax^2-2bx+c$, where a, b and c are constants with $a>0$. Given that the vertex of the ...
0
votes
1answer
36 views

Prove that $\{a_n\}$ is bounded [duplicate]

Let $\{a_n\}$ be a sequence an $L$ a real number such that $\lim_{n\to\infty} a_n = L$ Prove that $\{a_n\}$ is bounded This reminds me of the bounded monotone convergence theorem (BMCT) but in ...
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votes
1answer
50 views

Prove that $f(x)$ is continuous at $a$

Prove that $f(x)$ is continuous at $a$ $\iff$ for all sequences sequence $\{a_n\}$ with $\lim_{n\to\infty} a_n = a$, $\lim_{n\to\infty} f(a_n) = f(a)$ I have no idea how to start this. Should I use ...
0
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1answer
69 views

Two statements R and S are logically equivalent iff R $\iff$ S is a tautology.

How do prove the following statement: "Two statements R and S are logically equivalent iff R↔S is a tautology. without using a true table.Would I have to use cases? So far I have done so far is that ...
1
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3answers
175 views

Symmetric matrices and eigenvalues

If the eigenvalues of a symmetric matrix $A$ are greater than 0, show that $v^{\top}Av > 0$ for every $v \ne 0$ I am trying to prove this as follows: If $v$ is an eigenvector of $A$, then $Av ...
1
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4answers
52 views

How to use Cross Product Properites to do proof

How do I proceed with a proof for this question? Prove that: \begin{equation} (a \times b) \cdot (c \times d) = \begin{vmatrix} a \cdot c & b \cdot c \\ a \cdot d & b \cdot ...
0
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1answer
32 views

Proof the statements

Proof the statements below i)If $P(A)=0$ and $B$ is any event, then $A$ and $B$ are independents ii)If $P(A)=1$ and $B$ is any event, then $A$ and $B$ are independents iii)The events ...
0
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0answers
4 views

Boltzmann-Planck Entropy Formula

Hi guys I want to ask if anyones know the full proof of Boltzmann-Planck Entropy Formula S=klnW With Honors Dimitrios Gerontitis
4
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1answer
128 views

Is Alfred Tarski's Introduction to Logic still helpful for self study?

I am trying to setup a self study path to enhance my knowledge of mathematical logic. I haven't taken a logic course for a few years and my confidence on mathematical proofs is unnerving. I am ...
1
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1answer
46 views

Principle of well ordering

Every non-empty set $A\subset\mathbb{N}$ have a smallest element, i.e. an element $n_0\in A$ such that $n_0\leq n$ $\forall n\in\mathbb{A}$ Proof: Let $I_n=\{p\in\mathbb{N};p\leq n\}$ the set ...
1
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2answers
40 views

Logically Equivalance - Proofs

In terms of logical statements, is ($\exists$n $\in$ N)($\forall$ x $\in$ A)(nx >= 1) equal to ($\forall$x $\in$ A)($\exists$ n $\in$ N)(nx >= 1)? Also consider the following statements $\forall x ...
2
votes
2answers
54 views

Logical Statements - Proofs

Let $f$ be a real-valued function (a function with target space the set of reals). Let $P(x, M)$ stand for $|f(x)| \leq M $, let $N$ be the set of positive real numbers, and let $\mathbb{R}$ be the ...
0
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2answers
69 views

Is a proof by counterexample considered a proof by contradiction?

My question is already in the title. Let us look at some example. I would like to prove that a game $G(n,m,u)$ does not have a pure Nash equilibrium (PNE), for example. I did it like this: Suppose ...