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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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 ...
2
votes
1answer
36 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
35 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 ...
-3
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
votes
1answer
66 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
vote
3answers
171 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
vote
4answers
45 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
votes
1answer
31 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
votes
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
votes
1answer
112 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
vote
1answer
43 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
vote
2answers
39 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
52 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
votes
2answers
51 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 ...
0
votes
1answer
80 views

Seemingly simple logic question

I found this pleasant textbook on Proof Theory online and free: Introduction to Proofs, an Inquiry-Based approach To quote (page 9): 2.26 DEFINITION. A sequence $\langle x_0,x_1, . . . ...
1
vote
0answers
21 views

Rank of the sum of two rank 1 matrices, proof check

Claim: $(\forall u\in \mathbb{R}^2)$ $(\nexists(\delta,v)\in(\mathbb{R}, \mathbb{R}^2))$ such that $uu'+vv'=\delta \begin{pmatrix} 1 & 0\\0 & 0 \end{pmatrix}$. That is, for any vector $u$ of ...
0
votes
4answers
52 views

Logic, writing proof

i)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 $y=3$ ii)Suppose that $x$ and $y$ are real numbers. Prove that if $x^2y=2x+y$, ...
0
votes
0answers
42 views

Principle of infinite descent

I'm trying to prove the following: "It is not possible to have a sequence of natural numbers which is in infinite descent. (Hint: assume for sake of contradiction that you can find a sequence of ...
2
votes
4answers
85 views

Non-standard model of $Th(\mathbb{R})$ with the same cardinality of $\mathbb{R}$

Let $\mathfrak{R}= ⟨\mathbb{R},<,+,-,\cdot,0,1⟩$ be the standard model of $Th(\mathbb{R})$ in the language of ordered fields. I need to show that there exists a (non standard) model of ...
2
votes
2answers
70 views

Number Theory- Mathematical Proof

I am trying to prove a theorem in my textbook using another theorem. What I need to show that if a,b, and c are positive integers, show that the least positive integer linear combination of a and b ...
0
votes
2answers
25 views

Prove that for all $x$, $y$ in $\mathbb{R}$ there exist $z$, $g$ such that $x = z + g$, $y = z - g$

if I want to prove the following: $\forall x, y \in \mathbb{R}\,\,\,\exists\,\,z, g : x = z + g, y = z - g$ Can the resolution of the following system act as a proof: $\begin{cases} x = z + g\\ y ...
4
votes
2answers
32 views

Check proof of some simple inequality

Can you check please my proof of this inequality? It's all right?
3
votes
1answer
44 views

Prove equality of Fibonacci sequence

Let $u(n)$ — the Fibonacci sequence. Prove that $$u(1)^3+...+u(n)^3=\frac{ 1 }{ 10 } \left[ u(3n+2)+(-1)^{n+1}6u(n-1)+5 \right].$$ I suppose we need prove that equality by iduction in $n$. ...
1
vote
0answers
13 views

Doob's submartingale stopping theorem in the context of the submartingale problem

Let $$X^\omega_f (t, w) = f(w(t)) - f(w(t \wedge \tau)) - \frac{1}{2} \int_{t \wedge \tau}^t \Delta f(w(s))\, ds$$ be a $P^\tau_\omega$-submartingale. 1) Why Doob's submartingale stopping theorem ...
0
votes
1answer
37 views

Proving an Iff Statement

Suppose we had a function defined over the complex numbers: $ f(x)= \begin{cases} 1&\text{if } x\in\mathbb{R}\\ 0&\text{if } x\not\in\mathbb{R} \end{cases} $ And we are asked to prove that ...
3
votes
3answers
62 views

Is it necessary and sufficient that $6$ divides $n^2$ for the positive integer $n$ to be divisible by $6$? [closed]

As the title suggests, is it a necessary and sufficient that $6$ divides $n^2$ for the positive integer $n$ to be divisible by $6$? Like, I understand the dictionary definitions of necessary and ...
7
votes
3answers
432 views

Some trouble with the induction

Prove, that for any positive integer $n \geqslant 2$ we have the inequality $$ \frac{ 4^n }{ n+1 } < \frac{ (2n)! }{ (n!)^2 }.$$ For $n=2$ the inequality is true. Directly just take and ...
0
votes
2answers
50 views

On proving $(f^{-1})'(b) = \frac{1}{f'(a)}. $ where $b = f(a)$.

Could somebody kindly provide a proof or a reference to a proof of this fact: Let $ I $ be an open interval, and suppose that $ f: I \to \mathbb{R} $ is one-to-one and continuous on $ I $. If $ f ...
1
vote
2answers
44 views

Proof in set theory

Let $A,B,C$ -- subsets in some fixed set. Prove that $A \cap B \subseteq C$ iff $A \subseteq \overline{B} \cup C$. Have no ideas how to prove this. On the language of definitions we have $$x ...
4
votes
2answers
98 views

Proof of determinant formula

I have just started to learn how to construct proofs. That is, I am not really good at it (yet). In this thread I will work through a problem from my Linear Algebra textbook. First i will give you my ...
1
vote
2answers
63 views

Proving that $\hat{x} = x$ in least square

I am trying to show that if $Ax=b$ has a unique solution $x$, then the least square solution $\hat{x}$ is the exact one (i.e., $\hat{x} = x$). My attempt: We know that for $A x = b$ to have an exact ...
1
vote
3answers
161 views

Prove that there are infinity many tautologies.

For this question I think I am suppose to use proof by contradiction, but I need some hints on how to proceed with the proof. Always if someone can give me a brief explanation on how proof by ...
0
votes
0answers
11 views

Theorem implication/equivalence transitiveness in demonstrations

Suppose having three theorems $A, B, C$ that it's necessary to show being equivalent and having the following hypothesis: We know that $A \Leftrightarrow B$ and $B \Leftrightarrow C$. It would ...
2
votes
4answers
226 views

How do I know which of these are mathematical statements?

While reading this book called "How to Read and do Proofs" by Daniel Solow(Google) I found the following exercise at the end of the first chapter. So how do I know if something is a mathematical ...
1
vote
2answers
84 views

Prove that $\sin^{2}{\theta} + \cos^{2}{\theta} = 1.$

I believe that I have been able to prove that Prove $\sin^{2}{\theta} + \cos^{2}{\theta} = 1, \forall \theta,$ but I would like to ask if my proof is correct / valid.
2
votes
1answer
26 views

Show that $dim(X,\succeq)\leq |X^2|$ when $X$ is finite

I am trying to prove that when $(X,\succeq)$ is a finite preorder, the $dim(X,\succeq)\leq |X^2|$. Here's the full context (Exercise 11 (a)): My idea of resolution was to show that any set of ...
1
vote
2answers
31 views

Path and cycle length proof

How can we prove this proposition : Every graph G contains a path of length $ \delta(G)$ and a cycle of length at least $ \delta(G) + 1$, provided that $\delta(G) \geq 2 $. ($ \delta(G)$ is the ...
1
vote
1answer
65 views

prove that $p(n) := n^2 + n + c$ is not prime

The question is in MIT Mathematics for CS assignments but unfortunately there is no solutions. -> I do understand that it is false if we use $n = c$ or $n = c-1$ but cannot formally write it as ...
4
votes
5answers
100 views

Prove every integer is of the form $5k+r$ with $0\le r<5$

I have came across this question from my text book: Prove or disprove: any integer $n$ is of the form: $5k$, $5k + 1$, $5k + 2$, $5k + 3$ or $5k + 4$ for some integer $k$. I'm not sure what would be ...
0
votes
0answers
41 views

Comparison of books that teach proof techniques

I have to take discrete math and want to learn proof techniques both to get ahead in it as well as open up the possibility of understanding higher math. I've seen several books recommended such as How ...
1
vote
1answer
30 views

Primary decomposition of modules - uniqueness proof

Let $M$ be $A$-module, $A$ commutative ring, and $N$ submodule and let $$N=Q_1\cap\dots\cap Q_r=Q'_1\cap \dots \cap Q'_s$$ be reduced primary decompositions of $N$. Then $r=s$. The set of primes ...