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
votes
2answers
46 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
18 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
84 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 ...
1
vote
1answer
59 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 ...
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
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 ...
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 ...
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 ...
2
votes
3answers
32 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 ...
1
vote
1answer
30 views

Help to prove the condition that a right half-open interval is not empty

The right half-open interval is defined as: $[a,b) = \{x \in \mathbb{R}|a \le x \lt b\}$ I need to prove: $[a,b) \ne \emptyset \iff a<b$ My attempt: For $\Rightarrow$: $$\begin{align} ...
0
votes
1answer
48 views

Why are the different ways to write a universal statements equivalent?

Consider the following universal statements: $\forall a \in \mathbb{R}-\{0\}, a^2 > 0$ $\{a \in \mathbb{R} - \{0\}| a^2 > 0 \} = \mathbb{R}-\{0\}$ $a\in \mathbb{R}-\{0\} \Rightarrow a^2>0$ ...
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
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} ...
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 ...
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
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 ...
1
vote
2answers
82 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
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 ...
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 ...
-2
votes
1answer
39 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.
12
votes
2answers
297 views

Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.

Is my proof correct? I have made use of the fact isomorphism preserves order of elements, which I proved couple of exercises back. I am also interested in other ways of proving it. Is there a more ...
2
votes
3answers
187 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 ...
0
votes
2answers
43 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 $n$
32
votes
5answers
5k views

Are all prime numbers finite?

If we answer false, then there must be an infinite prime number. But infinity is not a number and we have a contradiction. If we answer true, then there must be a greatest prime number. But Euclid ...
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
32 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 = ...
30
votes
5answers
3k views

Is it okay to reverse engineer proofs in homework questions?

In a linear algebra text book, one homework question I received was: Prove that $\mathbf{a \cdot b} = \frac{1}{4}(\|\mathbf{a + b}\|^2 - \|\mathbf{a - b}\|^2)$. Where $\mathbf{a}$ and ...
2
votes
3answers
72 views

How to Prove This Trigonometry Identity?

I have to prove that: $$\tan^2\theta \sin^2\theta = \tan^2\theta - \sin^2 \theta$$ Here is what I have tried $$\tan^2\theta \sin^2\theta$$ ...
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. ...
5
votes
2answers
116 views

What is an instance of a mousetrap proof?

A part of the first chapter of the book The spirit and the uses of the mathematical sciences talks about the beauty of mathematics. The author quotes from a lecture of Hasse and introduces the notion ...
4
votes
1answer
38 views

Proving a Subset Identity

Working on part A of this problem: I worked out the first part like this: 1) If $A$ is a subset of $B$ then $\forall~x~[x\in A \implies x\in B]$ 2) Same goes for $C$ being a subset of $D$ (If ...
0
votes
3answers
28 views

Help with understanding definition of divisibility in this case.

I have a proof that shows that if $5 \mid xy$ then $5 \mid x$ or $5 \mid y$. It's pretty clear to me that I can just say that suppose $5 \mid x$, then $x=5a$, where $a$ is an integer. then $xy = ...
2
votes
5answers
67 views

Proof regarding divisibility

if $n\in\mathbb{Z}$, then $4$ does not divide $(n^2 - 3)$ I'm not sure how to approach this question, I know how to do questions that involve proving that it does divide but I'm unsure of how to do ...
0
votes
3answers
27 views

Help to prove $(A \times B)\cup (C \times D) \subseteq (A\cup C) \times (B\cup D)$

Prove $(A \times B)\cup (C \times D) \subseteq (A\cup C) \times (B\cup D)$ My attempt: $\begin{align} (x,y) \in (A \times B) \cup (C \times D) & \Rightarrow & (x,y) \in (A \times B) \vee ...
2
votes
0answers
14 views

Set with relative complement forms partition

Prove that if $S$ is a set and $ \emptyset \subsetneq A \subsetneq S $ then $\Pi = \{A , S-A \}$ is a partition of $S$. Proposed Solution: Since $ A \subsetneq S$ , we have $S - A \neq ...
-1
votes
3answers
86 views

Examples of revisited proofs after new theorems are discovered… [closed]

Are there any nice examples of "old" complicated proofs that become much simpler after new math is discovered years later? For instance, we know now that Pn+16< Pn+1 occurs infinitely often (where ...
4
votes
1answer
151 views

prove $x \mapsto x^2$ is continuous

I am to show the continuity of this function with the help of $\epsilon$-$\delta$ argument. The function is: $g: \Bbb{R} \rightarrow \Bbb{R}$, $x \mapsto x^2$. Given the $\epsilon$-$\delta$ ...
0
votes
1answer
26 views

Need help proving the statement

Assume that D ⊂ N and D ̸= ∅. Prove or disprove using a detailed structured proof, justifying every step: [∀x ∈ D, ∃y ∈ N, y < x] ⇔ [0 ̸∈ D] I have no idea how to prove a statement like that, I'm ...
2
votes
1answer
28 views

Question on Induction (Very Simple)

I've just started a course in mathematics at university, and our current topic is mathematical induction. I've been given the following question: $$1+4+4^2+....+4^{n-1}=\frac{4^{n}-1}{3}.$$ I get ...
1
vote
1answer
330 views

proof of validity of tautology in first order logic

Every first-order logic formula which has a tautological shape in propositional logic is a valid formula. Will it be possible to give a formal proof for the above ? Thanks and Regards.
2
votes
1answer
42 views

Proof Verification for Discrete Math Class

Prove that $n^2$ is even iff $n$ is even. I proved it like this: Case I: $n$ is even 1) $n = 2a$ $(a\in Z)$ 2) $n^2 = 4a^2 = 2(2a^2)$ 3) $2a^2 = K$ $(K \in Z)$ 4) $n^2 = 2K$ Case II: $n$ is ...
1
vote
1answer
34 views

Decomposition of a function in positive and negative parts and its integrability

Is it true to say that $\int_\mathbb{R}|f(x)|dx<\infty\Rightarrow\int_\mathbb{R}f(x)=0$?
7
votes
3answers
89 views

How to structure long proofs

How do you structure proofs that are longer than say half a page? I have already encountered a variety of styles (in my short math life), some of which I list below and I just hoped to hear some wise ...
0
votes
2answers
27 views

Proof for combination using a specific definition

Suppose $n,k,\in\mathbb{Z}$ and $0\leq k \leq n $ prove using the following definition: if n and k are integers then $\binom{n}{k}$ denotes the number of subsets that can be made by choosing k ...