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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2answers
46 views

Construction of Natural Numbers

I am trying to prove that the natural numbers can be constructed from the product of a power of $2$ and an odd number. For all $n \neq 0$ in the natural numbers, $n = (2k+1)(2^p)$, where $k$ and $p$ ...
1
vote
2answers
18 views

Prove using PMI that if $A$ is denumerable and $B$ is finite, then $A \cup B$ is denumerable.

This is what I have thus far: Claim: If $A$ is denumerable and $B$ is finite, then $A \cup B$ is denumerable. Proof. Suppose $A$ is denumerable and $B$ has $n$ elements and $B = \{b_1, b_2, b_3, ...
0
votes
1answer
10 views

Question about choosing cases in a proof by cases

Prove that for every integer $n \ge 8$, there exist nonnegative integers $a$ and $b$ such that $n = 3a + 5b$. Proof: Let $n \in \mathbb Z$ with $n \ge 8$. Then $n = 3q$ where $q \ge 3, n = 3q + ...
3
votes
1answer
82 views

Show that $\bar A = A \cup [(0,0), (0,1)]$

In $(\mathbb R^2, ||\cdot||_{\infty})$, let: $A_0 = ]0,1] \times \{0\}$ $A_n = [(\frac{1}{n}, 0), (\frac{1}{n}, 1)]$ for each $n \ge 1$. $A = \cup_{n=0}^{\infty} A_n$ It is required to prove that: ...
-1
votes
1answer
34 views

Function is identically zero almost everywhere

Prove that if $\int_E f d\mu = 0$ for some $f \ge 0$, then $f = 0$ almost everywhere. This is Execrise 1 in Chapter 11 of baby Rudin. My attempt: $\int_E f d\mu = 0 \implies$ sup { ${\int_E s ...
1
vote
2answers
23 views

shown that f is even

lets one function $f:\mathbb{R}-\{0\}\mapsto\mathbb{R}$ where $\mathbb{R}$ is the set of reals, lets $f$ such that $f\left(\frac{a}{b}\right)=f(a)-f(b)$ for every $a$ and $b$ in belonging to the ...
2
votes
1answer
42 views

Satisfiability proof of formulas with pure literals

Let $\varphi$ be any propositional formula in negation normal form (NNF). A literal $\ell$ is pure in a formula $\varphi$, if the complement of $\ell$, $\ell^c$, does not occur in $\varphi$, where ...
9
votes
3answers
1k views

Prove that there is no smallest positive real number

I have to prove the following: $$\text{Prove that there is no smallest positive real number}$$ Argument by contradiction Suppose there is a smallest positive real number. Let $x$ be the smallest ...
1
vote
2answers
28 views

Which of the properties, Reflexive, Irreflexive, Symmetric, Asymmetric, Antisymmetric, Transitive, Linear, does F satisfy?

Let $S={(n,m) ∶n,m∈Z^+}$. Define the relation F on S by ${(n,m),(i,j)}∈F$ if and only if $nj=mi$. In other words, let $F = {((n, m), (i, j)) ∈ S × S: nj = mi}$. Proof F is reflexive: Show that for ...
3
votes
1answer
23 views

Elementary question on set theory

Suppose $A \subset B$ then does this imply $B^{c} \subset A^{c}$? Here, $B^{c}$ denotes the complement of $B$. I have tried drawing Venn Diagrams and it seems obvious but is there a formal rigorous ...
4
votes
0answers
28 views

Proving $f$ cannot be onto

If $f$ maps finite sets $A$ to $B$ and $n(A) < n(B)$, prove that $f$ cannot be onto. Proof by contradiction: If $f: A→B$ and $n(A) < n(B)$, $f$ is onto. Since, by definition of a ...
0
votes
3answers
54 views

Prove that $A$ is countable.

Hi so I'm practicing for a exam and I need help to figure this proof out, Suppose $A\subseteq \mathbb R^+$, $b\in\mathbb R^+$, and for every list $a_1,a_2,\ldots,a_n$ of finitely many distinct ...
4
votes
3answers
41 views

Prove A is an open set if and only if $A \cap Bd(A) = \emptyset $

Prove A is an open set if and only if $A \cap Bd(A) = \emptyset $ Here is my start: Suppose A is an open set. We know $X-A$ is closed. Need to show $A \cap Bd(A) = \emptyset$ Let $ x \in A$. ...
0
votes
1answer
31 views

Prove that, in the usual topology, both $a$ and $b$ are in the boundary of each $(a,b)$ and that no other point is in the boundary.

Suppose that $a$ and $b$ are real numbers such that $ a \lt b$. Prove that, in the usual topology, both $a$ and $b$ are in the boundary of each $(a,b), [a,b], [a,b),$ and $(a, b]$, and that no other ...
2
votes
2answers
36 views

Prove that Set B is countable - Is this proof correct?

It seems that I have some issues with the rigor of this proof and I don't know what I'm doing wrong. Could someone tell me if this proof is correct and rigorous enough? Here's the question Prove ...
0
votes
1answer
41 views

what's the answer for this proof [on hold]

What's the answer for this question: Show that: a) a−∅ = a. b) ∅−a = ∅. I am already try to solve this, but I feel this is not logical solution. a) A−∅ = A A−∅ ={x|x ∈A^ ...
0
votes
1answer
32 views

Extending the transitive property [on hold]

Suppose we have a transitive relation $R$ on a set $S$. Suppose for some $n\in\mathbb{Z}^+\colon (s_0, s_1),(s_1,s_2),\ldots,(s_{n-1}, s_n)\in R$. Show that: $(s_0, s_n) \in R$ So I am having ...
9
votes
2answers
3k views

Proving rigorously the supremum of a set

Suppose $A \subset \mathbb{R} \neq \emptyset$. Let $A = [\,0,2).\,\,$ Prove that $\sup A = 2$ This is my attempt: $A$ is the half open interval $[\,0,2)$ and so all the $x_i \in A$ look like $0 ...
1
vote
4answers
64 views

Solution verification: Prove by induction that $a_1 = \sqrt{2} , a_{n+1} = \sqrt{2 + a_n} $ is increasing and bounded by $2$

I have the following recursive relation (sequence): \begin{align} a_1 = \sqrt{2}, \quad a_{n+1} = \sqrt{2 + a_n} \end{align} My Try: I'm a little skeptical of my manipulations near the end but it ...
0
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0answers
35 views

How to prove this set P is countable? [duplicate]

Hi so I'm a beginner to proofs and these day's I'm studying infinite sets. I'm trying to figure out the proof for the following: Let P = {X$\in \mathscr{P}({\mathbb{Z}}^+)$| X is finite}. Prove ...
0
votes
2answers
55 views

Power set equinumerosity. Is this proof correct?

So I'm trying to prove the following, Prove that if $A\sim B$ then $\mathscr{P}(A) \sim \mathscr{P}(B)$. Here's how I started out to prove there is a function that is injective: Suppose $A ...
1
vote
0answers
39 views

Hilbert Space is not locally compact.

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. Show that Hilbert Space is not locally compact at any point. This is what I understand: ...
2
votes
1answer
44 views

Proving an Inequality (terms won't cancel out)

Problem: If $x$ and $y$ are real numbers such that $y \geq 0$ and $y(y+1) \leq (x+1)^2$, prove that $y(y-1) \leq x^2$. This is what I tried: \begin{align} y(y+1) \leq (x+1)^2 &\implies y^2 + y ...
2
votes
0answers
23 views

Product Spaces: Tube Lemma

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. My professor asked to prove the following Lemma. The Tube Lemma: Let $K$ be a compact ...
0
votes
1answer
41 views

Proof that polynom $P:\mathbb{R}^n\to\mathbb{R}$ is continuous

Could you tell me some webpages or books where I can find the proof that polynom $P:\mathbb{R}^n\to\mathbb{R}$ is continuous. I know how it can proof if $P:\mathbb{R}\to\mathbb{R}$, but I don't know ...
1
vote
1answer
55 views

What should I learn to increase my skill to find proof?

I know... reading lot of proofs and comments about them and working hard by myself on proving theorems are probably the only good solutions. But in the same time, it is not a solution at all because ...
0
votes
1answer
39 views

I am trying to use proof of sequence correctly to make valid

Here I am trying to use a proof sequence so that the argument is valid (hint: the last A’ has to be inferred). (A → C) ∧ (C → B') ∧ B → A' Here are my steps I tried but not sure if this is correct ...
1
vote
0answers
38 views

Uncountability of $\mathbb{R}^I$ if $I$ is uncountable

Prove that if $I$ is uncountable, then $\mathbb{R}^I$ with the product topology is not countable. Based on what I have read, a set is uncountable if there is a bijection from that set to ...
1
vote
4answers
67 views

Prove that for all positive integers $n$, $2^1+2^2+2^3+…+2^n=2^{n+1}-2$ [duplicate]

I want to prove that for all positive integers $n$, $2^1+2^2+2^3+...+2^n=2^{n+1}-2$. By mathematical induction: 1) it holds for $n=1$, since $2^1=2^2-2=4-2=2$ 2) if $2+2^2+2^3+...+2^n=2^{n+1}-2$, ...
0
votes
3answers
42 views

How to show $f(x) = x^2 + x + 1$ is continuous? [closed]

Using the $\epsilon-\delta$ definition of continuity to prove that a particular function is continuous: Let $f(x) = x^2 + x + 1$. Given a positive number $\epsilon$, find a positive $\delta$ such that ...
0
votes
2answers
61 views

Explaining why proof by induction works [duplicate]

I am learning math proofs for the first time. So I cannot discern the reason for all the details in a proof. Here's the statement of mathematical induction: For every positive integer $n$, let ...
1
vote
2answers
25 views

Proof by induction of the Inequality of Harmonic numbers: $H_{2^n} \ge 1+ \frac n2$

My question is, for the question below, in the inductive step, where does $\dfrac{1}{2^{(k+1)}}$ come from?And where does $2^k$ come from in the third last step?
3
votes
1answer
27 views

Uniqueness Proof procedure

I'm reading a book on understanding math proofs to enable me to understand mathematics at a deeper level. Along the way I came across this: An element belonging to some prescribed set $A$ and ...
0
votes
0answers
46 views

Trying to justify each step correctly in proof sequence

I am trying Justify each step in the proof sequence below for correctly with [A → (B ∨ C)] ∧ B' ∧ C' → A' So I justified my steps here but I am not sure at 1 to 3 if I did it correctly. A → (B ∨ ...
107
votes
10answers
6k views

Why do people use “it is easy to prove”?

Math is not generally what I am doing, but I have to read some literature and articles in dynamic systems and complexity theory. What I noticed is that authors tend to use (quite frequently) the ...
3
votes
1answer
53 views

Let $a$, $b$, and $c$ be elements of a commutative ring, and suppose $a$ is a unit. Prove that $b$ divides $c$ if and only if $ab$ divides $c$.

Let $a$, $b$, and $c$ be elements of a commutative ring $R$, and suppose $a$ is a unit. Prove that $b$ divides $c$ if and only if $ab$ divides $c$. Okay so here is the proof I came up with. Please be ...
2
votes
4answers
392 views

How do I write this proof formally?

How can I formally prove that $$\max\{\lvert x+y\rvert _i \} \leq \max\{ \lvert x_j \rvert \} + \max\{\lvert y_k \rvert \}$$ Where $x,y$ are the components of a $n$-vector with $1 \leq i,j,k \leq ...
1
vote
1answer
40 views

Formal language: Proving the reverse operation on a word through induction

I'm practicing proofs and given the following statement: Let $\Sigma$ be an alphabet, $\epsilon$ the empty word and $\sigma:\Sigma^{*}\rightarrow\Sigma^{*}$ an operation which for $a\in\Sigma$ and ...
0
votes
1answer
38 views

How to prove this theorem? (Logical symbols help)

This is a theorem from a book. I'm having a hard time on proving it. Suppose A is a set,$\mathcal{F}\subseteq \mathscr{P}(A)$, and $\mathcal{F} \neq \emptyset$. Then the least upper bound of ...
0
votes
2answers
156 views

Proving $x^{2}+1 \neq n! $,using Gaussian Integer.

I want to show that $$x^{2}+1 \neq n! $$ for $n>3$ where $x,n$ are both integers. Since $$x^{2}+1=(x+i)(x-i) $$ it follows that $x^{2}+1$ has only prime factors on the form ($4k+1$), whereas $n!$ ...
0
votes
3answers
159 views

Proof of the Dirichlet–Dini Criterion for Pointwise convergence of Fourier series

I have tried and failed to prove the Dirichlet–Dini Criterion for Pointwise convergence of Fourier series which is as follows (and is described here: ...
4
votes
1answer
42 views

Introduction to proofs. [duplicate]

I am not at all familiar with mathematical proof-writing and would like to learn how to create my own proofs. So, I was wondering whether it would be possible for you to recommend me to any book or ...
1
vote
2answers
43 views

Summation Proof Dealing With 3s Multiples [duplicate]

So the problem is as follows: Prove that if the sum of digits of a decimal $n$ is three's multiple, then n is three's multiple by direct proof. For example, $11234567$ is 3's multiple because ...
1
vote
2answers
38 views

Proving $10^n \equiv 1 \pmod 3$ for all $n\geq 1$ by induction

Prove that $10^n \equiv 1 \pmod 3$ for all positive integers $n$ by mathematical induction. Can someone please help me in solving this problem and explain what's going on? Any guidance would be ...
-2
votes
0answers
53 views

Summation Direct Proof Help [closed]

Prove that if the sum of digits of a decimal n is three's multiple, then n is three's multiple by direct proof. For example, 11234567 is 3's multiple because 1+1+2+3+4+5+6+7=24, and in fact, 11234567 ...
3
votes
1answer
17 views

Let $R$ be any rotation and $P$ any reflection then $R \circ P$ and $P \circ R$ are both glide reflections

Let $R$ be any rotation and $P$ any reflection then $R \circ P$ and $P \circ R$ are both glide reflections I am having trouble showing $P \circ R$ is a glide reflection, I manage to get $R \circ P$, ...
1
vote
1answer
59 views

Proving that the gamma function is a certain limit

This time I want to prove that $\displaystyle \Gamma(x) = \lim_{\varepsilon\rightarrow 0} \int_\varepsilon ^{1/\varepsilon} t^{x-1} e^{-t}$, I know this is true because we have defined $\displaystyle ...
-2
votes
0answers
40 views

Why does this proof by bashing not work?

Given is a triangle $ABC$ with circumcenter $O$ and orthocenter $H$. The feet of the perpendiculars from $C$, $B$, and $A$ to the opposite sides are $F$, $E$, and $D$ respectively. Prove that ...
0
votes
1answer
14 views

Using existential instantiation on a universally quantified given

I'm trying to prove the following exercise of How to Prove it: A structured Approach (Section 3.4, exercise 19): Suppose A, B and C are sets. Prove that A $\triangle$ B and C are disjoint iff A ...
2
votes
1answer
605 views

Proving if the limit of $f(x)$ approaches zero, then the limit of $1/|f(x)|$ approaches infinity.

The Problem: Suppose that $f:D\to\Bbb R$, where $D$ is a subset of $\Bbb R$ and $a$ is an accumulation point of $D$, $\lim_{x\to a}f(x)=0$, and $f(x)\ne0$ for any $x$ in $D$ in some neighborhood of ...