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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0
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3answers
52 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
46 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
90 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
44 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
45 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
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0answers
23 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
27 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
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2answers
62 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
36 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
16 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
57 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
43 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
57 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
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4answers
36 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 ...
0
votes
0answers
43 views

What does “Ad” mean in a proof? [closed]

My prof tend to use the word Ad. when writing proofs. Can someone please explain what "Ad." mean in a proof?
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
30 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
114 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
52 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 ...
1
vote
4answers
67 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
41 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
38 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
85 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 ...
1
vote
0answers
34 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
67 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
vote
2answers
62 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
votes
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
28 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
44 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
67 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
23 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
votes
2answers
35 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
vote
2answers
39 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
vote
2answers
40 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
vote
1answer
12 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
16 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
131 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
votes
2answers
51 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
20 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
29 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 ...