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.

learn more… | top users | synonyms

0
votes
2answers
34 views

Proving that any common multiplication of two numbers is a multiplication of their least common multiplication

Im trying to prove that if there are to numbers $n,m$ (natural numbers), and their smallest common multipe is $k$, so that $k = n·i$ and $k = m·j$ for some $i,j$ natural numbers, any common multiple ...
1
vote
1answer
34 views

Strange proposition in probability book for conditional probability

I found the following proposition (15.1) in the probability book of Heinz Bauer: Let us given that $X$ is a numeric random variable on $(\Omega,\mathcal{A},P)$ which is non-negative / ...
1
vote
1answer
22 views

Proving the properties of big union of unions for indexed sets

Let $I$ be an index set, and for each $i \in I$, let $J_{i}$, be another index set. For each $i \in I$ and $j \in J_{i}$, let $U_{j}$ be a set. Set X = $\bigcup\limits_{i\in I}J_{i}$. Prove that: ...
1
vote
1answer
26 views

Proving properties of closures using intersection of indexed sets and topology

How would I write a proof for this example? Let X be a set, and let $B \subseteq \mathcal P \left({X}\right)$. Define $B^* =${ $U \subseteq X:$ There is an index set $I$ and $U_{i} \in B$ for each ...
0
votes
3answers
33 views

How do I prove $x^n < x^m$ when $m > n$ and $x > 1$

Title I made an attempt at it here: $x^n < x^m$ when $m > n$ and $x > 1$, $m$ and $n$ are naturals so divide both sides by $x^n$ so $1 < x^{m-n}$ but here i am stuck. Please help!
0
votes
2answers
35 views

Prove that the order of $U(n)$ is even when $n>2$.

I'm trying to provide a solution to the following claim: "The order of $U(n)$ is even when $n>2$." Note: here, $U(n)$ is the set of all positive elements that are less than and relatively prime ...
0
votes
1answer
32 views

Question about proofs with topological spaces

How would I write a proof for this example? Let $(X, \tau_{1})$, $(Y, \tau_{2})$ and $(Z, \tau_{3})$ be topological spaces. A function ${f}: X \rightarrow Y$ is said to be continuous if for every V ...
0
votes
1answer
37 views

Question about writing proofs for limit

I intuitively understand proof with limits, but I'm not sure on how to write a formal proof for this example. For each $n \in \mathbb{N}$, let $a_n$, $b_n$ be real numbers. Also, let $a_{\infty}$, ...
0
votes
0answers
30 views

Question about proving a real number

If we know that for any, $\alpha \in \{0, 2\}^\mathbb{N}$ that $0 \le \sum\limits_{k = 0}^\infty\frac{\alpha(k)}{3^k} \le 3$, then what property of real numbers do we have to use to prove that ...
0
votes
1answer
65 views

Quick question about $\epsilon -\delta$ proofs

There is one step in $\epsilon - \delta$ proofs that I hope somebody could bring clarity to for me. Say we wanted to show $\displaystyle \lim_{x \to 2} x^2 = 4 $. Somewhere along the proof we would ...
1
vote
1answer
30 views

Question about written proof for geometric summation

Suppose $\alpha$ $\ne$ $\beta$ $\in \{0, 2\}^\mathbb{N}$ Prove that $$\sum\limits_{k = 0}^\infty\frac{\alpha(k)}{3^k} \ne \sum\limits_{k = 0}^\infty\frac{\beta(k)}{3^k}. $$ This is the written proof ...
-1
votes
1answer
27 views

Question about proof with the real numbers [on hold]

I understand that because $\{0, 2\}^\mathbb{N}$ has the same cardinality as $2^\mathbb{N}$, you can use this to prove an injection. However, I'm not sure on how to write a proof to relate $\mathbb{R}$ ...
0
votes
1answer
41 views

Help to clarify inductive step for proof of $\mathbb{N_m} \rightarrow \mathbb{N}_n\Rightarrow m\le n$

The statement to prove is: If there exists an injection $\mathbb{N_m} \rightarrow \mathbb{N}_n$ then $m\le n$ The solution says to prove by induction on $n$. I just need help on the inductive ...
0
votes
1answer
41 views

How to prove that a given map is an injection?

Let $g:\mathbb{N_{m_1-1}}\rightarrow \mathbb{N}_{m_1}$, where: $$g(i) = \left\{ \begin{align} i & \text {, for } i<i_0 \\ i+1 & \text{, for } i \ge i_0 \end{align}\right.$$ and $i_0 ...
0
votes
5answers
48 views

Proof of $\forall n \in \Bbb N$, $n > 2 \implies n! < n^n$

What I've got so far is this: Base case: n = 3 then $3 *2 * 1 = 6$ and $3^3 = 27$ $\therefore 6 < 27, 3! < 3^3$ So the base case is true. So if we assume $n! < n^n$ (n > 2) $(n + 1)! = ...
1
vote
2answers
26 views

If $R$ is a transitive realation, then $R\circ R\subseteq R$

Here's the question I'm struggling with: Let R be a transitive relation on a set A. Prove the R composed with R is a subset of R. I'm kind of lost on how to prove this. I've started with saying: ...
1
vote
4answers
41 views

Prove $\forall n \in\Bbb N$, $0 < a < 1$ $\implies$ $a^n \leq 1$

I'm trying to prove this by induction but I'm running into some trouble. The base case is $0$, so, $a^0 = 1$, the inequality holds true Being new to induction, I don't exactly know what to do for ...
-1
votes
2answers
40 views
2
votes
1answer
46 views

Question about proof with geometric sums

I am confused on how to write proofs for geometric sums. I think that using the well ordering principle to find the least n $\in$ $\mathbb{N}$ with $\alpha(n)$ $\ne$ $\beta(n)$ would be a good ...
0
votes
1answer
45 views

Question about proofs with limits

I intuitively understand proof with limits, but I'm not sure on how to write a formal proof for this example. For each n $\in$ $\mathbb{N}$, let $a_{n}$ be a real number. Also, let $a_{\infty}$ be a ...
0
votes
2answers
39 views

How do I reduce this induction problem at the k+1 step

Show that $2^n > n^2$ through induction and so far I got to the $k+1$ step, but I am stuck. I have $2^{k+1} = 2 +2^k$, but I don`t know how the book turned it into $k^2 +k^2$. The book then ...
0
votes
2answers
50 views

Proof by induction $\sum_{k=1}^{n}$ $k \binom{n}{k}$ $= n2^{n-1}$ for each natural number n [duplicate]

Prove by induction that $\sum_{k=1}^{n}$ $k \binom{n}{k}$ $= n2^{n-1}$ for each natural number n
0
votes
0answers
19 views

Orthonormality and fourier transform

If $g\in\mathcal{L}^2(\mathbb{R})$ then $\sum_{k\in\mathbb{Z}} |\hat{g}(\zeta+2k\pi)|^2=1$ for a.e $\zeta\in \mathbb{R} \Rightarrow \{g(.-k): k\in \mathbb{Z}\}$ is an orthonormal system. Please ...
1
vote
2answers
86 views

Prove: If $A \subseteq B$ and $B \subseteq C$, then $A \subseteq C$.

Can someone tell me if this proof is acceptable? Assume $A \not\subseteq C$. This means that there is an $x \in A$ such that $x \not\in C$. But since $\forall x \in A: x \in B$ and $\forall x \in B: ...
0
votes
1answer
61 views

Prove the reflexivity of $\subseteq$.

My professor gave me a list of exercises, I've been able to figure out what mechanism I should exploit to prove them, but I'd like to know if it's good. Until now we've been taught a little logic and ...
0
votes
2answers
23 views

Proof exercise: finding hypothesis and conclusion in a statement

I am starting learn mathematical proofs and I was doing some exercise that needed to identify the hypothesis and the conclusion in a given statement. And I'm having trouble trying to figure it out in ...
0
votes
3answers
25 views

Can the proof for the following 4 cases be simplified to 2 cases?

Let $X$ and $Y$ be finite and disjoint sets. Suppose we are required to prove the following: $|X|\ge 0 \text{ and } |Y|\ge 0 \Rightarrow Q $ where $Q$ is some statement. Therefore, I know I need to ...
1
vote
1answer
60 views

How to prove that $f:\mathbb{N}\rightarrow X$ where $f$ maps to an element in a set, is a bijection?

Let $X$ and $Y$ be disjoint finite sets, $|X|=n$ and $|Y|=m$, so that we have the following bijections: $f:\mathbb{N}_n \rightarrow X$ and $g:\mathbb{N}_m \rightarrow Y$ I need to prove that ...
1
vote
4answers
50 views

Direct proof using modular arithmetic

Give a direct proof of $8\mid (3^n + 5^n)$ for all odd natural numbers. I know how to prove this by induction, I am not sure how to go about it using a direct proof. I would start by saying that ...
2
votes
3answers
34 views

Proof involving lcm and biconditional statement.

Suppose $a,b\in\mathbb{Z}$. Then $a = \operatorname{lcm}(a,b)$ if and only if $b\mid a$ Unsure of how to approach this problem.
2
votes
4answers
87 views

Prove $A \subset \emptyset \iff A = \emptyset$

How does one prove this? Can one prove by contradiction by saying: Let $A$ be any set such that $A$ contains at least one element. Now assume $A \subset \emptyset$. This is clearly absurd by the ...
0
votes
3answers
33 views

Proof for a non-conditional statement

I'm having a bit of trouble doing this proof. If $a\in\mathbb{Z}$, then $a^3 \equiv a \pmod 3$. I know how to do proofs if there were conditional statements but not sure how to prove this with ...
0
votes
2answers
51 views

If a converse of an implication is false, does this mean that the proof of that implication will always have an implication that is not reversible?

Let $f:X \rightarrow Y$ be a function and $B_1, B_2 \in \mathcal{P}(Y)$. Prove that $B_1 \subseteq B_2 \Rightarrow \overleftarrow{f}(B_1) \subseteq\overleftarrow{f}(B_2)$. My attempt: $\begin{align} ...
2
votes
2answers
30 views

Help to prove $f$ is surjective $\Leftrightarrow \forall y \in Y, (X \times \{y\} \cap G_f ) \ne \emptyset $

Let $f:X \rightarrow Y$ be a function with graph $G_f \subseteq X \times Y$. Prove that $f$ is surjective if and only if $\forall y \in Y, (X \times \{y\} \cap G_f ) \ne \emptyset $ I have some ...
0
votes
1answer
24 views

Prove this statement (inequality)

$x,y,z$ $\in\mathbb R$ then $|x-y| \leq|x-z|+|y-z|$ Prove this statement. I thought it was the triangle inequality, but I can't seem to end up with the correct order.
1
vote
1answer
34 views

Correctly understanding truth table problem?

I'm typing up a solution set for an "intro to proof" course. One of the problems asks the student to "construct a truth table for $(P \implies Q) \implies (\neg P)$." I interpreted this as requesting ...
1
vote
3answers
31 views

Very simple proof help about integers

I was wondering if anybody could help me with proving something very simple: that $9n\ne6$ when n is any integer. It seems extremely intuitive but I don't know how to make it into part of a rigorous ...
0
votes
1answer
36 views

Understand step in computing marginal distribution of restricted Boltzmann Distribution

Proof taken from http://image.diku.dk/igel/paper/AItRBM-proof.pdf (page 24) I understand everything up to and including: (1) $$p(\textbf{v}) = \frac{1}{Z}e^{\sum_{j=1}^mb_jv_j} \prod_{i=1}^n\sum ...
1
vote
0answers
57 views

Prove: $a^m\cdot a^n \cdot a^p=a^{m+n+p}$

How can I prove the following: Prop.: let be $m,n,p \in \Bbb{N}$ and $a \in \Bbb{R}$ then $$a^m\cdot a^n \cdot a^p=a^{m+n+p}$$ ??? I thinked by induction and I must prove: 1) $a^0\cdot a^0 \cdot ...
1
vote
2answers
38 views

Invertible Proof with transposed matrices

Let A, B, C be square matrices that are invertible. Say I want to express X with no inverses Say $$ (A^{T}A)^{-1}(X +B^ {T})(C^{-1}B^{-1})^{T} = I. $$ I know that $A^{T}A$ = $I$, but where can I go ...
-2
votes
1answer
41 views

Prove that every finite A is Dedekind finite. [duplicate]

I'm trying to prove that every finite set $A$ is Dedekind finite. I have to use the theorems: that a set $A$ is finite iff there is a natural number $n$ so that there is a bijection $f: n \rightarrow ...
0
votes
1answer
50 views

Prove that every positive natural number is Dedekind finite.

Prove that every positive natural number is Dedekind finite. I'm trying to prove this theorem with induction. I'm stuck on how I should use induction to prove this theorem.
0
votes
0answers
31 views

Trying to prove that two angles are congruent in a isosceles trapezoid

I was given this assignment to do the following. Write a paragraph proof for the following scenario. Given: KLMN is an isosceles trapezoid. Prove: ∠LKM is congruent to ∠MNL The thing is that I ...
0
votes
2answers
44 views

Prove that $n<(3/2)^n$ for any $n$ with induction [closed]

need help with induction with inequality, I suck at it. $n<\left(\frac{3}{2}\right)^n$ for any $n$
2
votes
3answers
188 views

Wheel of Fortune Problem

The Summation formula is $$\sum_{i=1}^ni =\frac{n(n+1)}2$$ How is it that we know the integers $1,2,...36$ appear exactly $3$ times. And why do we multiply the sum by $3$ in the last part of the ...
2
votes
2answers
73 views

Proving Lebesgue integration result

I have a Lebesgue integration question and a proposed proof. Please advise. Let $\Omega \subset \mathbb{R}^{n}$(denote the boundary as $\partial \Omega$) and consider $$\int_{\partial \Omega} vf ...
1
vote
2answers
33 views

Proof on limit superior and limit inferior of a set

I understand the result intuitively but how can I prove this? For a given integral $n \ge 1$, let $A_n = \left\{\frac mn \mid m \in \mathbb Z\right\}$. Show that $\varlimsup_{n\to\infty} A_n = ...
0
votes
2answers
59 views

Proving Alternating Series Convergence

Suppose $x_n > 0$ and $\sum_{n=0}^\infty x_n$ is convergent. Prove that $\sum_{n=0}^\infty (-1)^nx_n$ is convergent. Any hints or starting points? So far I figured that I should show that the ...
1
vote
2answers
85 views

Questions about Proof of Lusin's Theorem

I am reviewing my analysis notes, and having trouble understanding certain parts of the proof to Lusin's theorem. $\textbf{Lusin's Theorem}$: Let $F: [0,1] \rightarrow [0,\infty)$ be a nonnegative, ...
1
vote
2answers
52 views

Wondering if proof is proper

so I have been working on learning some new math in order to prepare for next year. I have been trying to learn proofs, and doing practice questions however the only problem is there are not answers. ...