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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1
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2answers
19 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
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 $p$ ...
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 + ...
-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 ...
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 ...
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 ...
1
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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 ...
3
votes
1answer
83 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: ...
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 ...
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^ ...
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 ...
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
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 ...
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: ...
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
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 ...
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
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 ...
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 ∨ ...
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 ...
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 ...
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 ...
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 ...
-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 ...
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 ...
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 ...
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 ...
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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 ...
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 ...
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 ...
1
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0answers
19 views

Is there a standard notation for $(p_i-k)(p_{i-1}-k)(p_{i-2}-k)\cdots$ where $k$ is a small positive integer

For $k=0$, there is: $p_i\# = (p_i)(p_{i-1})(p_{i-2})\cdots(5)(3)(2)$ For $k=1$, there is: $\varphi(p_i\#) = (p_i-1)(p_{i-1}-1)(p_{i-2}-1)\cdots(5-1)(3-1)(2-1)$ Is there any other notation that ...
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 $R \circ P$, ...
2
votes
0answers
23 views

Proof: The inverse of the translation $T_{AB}$ is the translation $T_{BA}$

Prove: The inverse of the translation $T_{AB}$ is the translation $T_{BA}$ This is my work so far: Let P be any point of the plane and set: $P'=T_{AB} (P)$ We want to show ...
0
votes
2answers
49 views

Prove or disprove that T:[0,2π] -> [0,2π] given by Tx = sin(2014x) is a contraction

i know that if we assume $T:[a,b] \to [a,b] $ and if $|T'(x)| ≤ α \space \forall \space a≤x≤b$ then T is a contraction . but unsure of how to apply that to this question
0
votes
0answers
54 views

Prove that $\int_{a}^{b} f(x)g'(x) dx = 0$ iff $f$ is constant

Given that $f$ is continuously differentiable and increasing on $[a, b]$, $g$ is differentiable on $[a, b]$, and $g'$ integrable on $[a, b]$. If $g$ is positive and $g(a) = g(b) = 0$, show that ...
0
votes
0answers
13 views

Upper bounds and Lower bounds (Relations Proof Problem)

So I've only recently started studying proofs and I've been using Velleman's "How to Prove it" This is a theorem from the book. I'm having a hard time on proving it. Suppose A is a ...
0
votes
0answers
36 views

Prob. 2.7-10 in Kreyszig's Functional Analysis Book: Is my solution good enough for anciliary purposes?

With valuable help from the SE community, I've managed to come up with the following solution to Prob. 10 after Sec. 2.7 in Introductory Functional Analysis With Applications by Erwine Kreyszig. I ...