For questions about the formulation of a proof, not about the mathematics behind it.

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0
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2answers
44 views

Can valid operations applied on an incorrect premises ever lead to true conclusions?

When we do a proof by contradiction, we assume a premise, derive a result from it, and if the result is incorrect then we conclude that our initial premise was wrong. However, if we do get a correct ...
4
votes
1answer
57 views

$G$ is abelian when any two non-identity $a$ , $b$ there is an automorphism $\delta$ such that $\delta(a)=b.$

$G$ is a finite group with identity $\mathcal e.$ Suppose for any two non-identity elements $a$ , $b$ of $G$ , there is an automorphism $\delta$ such that $\delta(a)=b.$ Then prove that $G$ is ...
0
votes
3answers
48 views

Show that $g$ is injective

How can one show that if $g(f(x))$ is injective and $f$ is surjective then $g$ is injective? Here is my attempt: $g(f(x))$ is injective, so $$g(f(a))=g(f(b)) \iff f(a)=f(b).$$ $f$ is surjective, ...
4
votes
0answers
54 views

Proof of inequality involving binomial coefficients

Proving things is not my forte. Stumbled upon following identity: $$\binom{n+k+1}{k}>\sum_{i=1}^k \binom{\alpha_i}{i}$$ For $\alpha_i<\alpha_j $ for $i<j$ $0\leq \alpha_i \leq n+k$ Also ...
0
votes
2answers
67 views

$({\mathbb{Q}},+)$ is not finitely generated

I'm trying to prove that $G = ({\mathbb{Q}},+)$ is not finitely generated. I have come up with the following, and would like to check it is correct: $G$ is generated by $\{1/n | n \in ...
0
votes
1answer
59 views

Is my proof valid for $9$ dividing sum of three consecutive cubes?

I am trying to use induction. Have I applied it correctly / rigorously enough? Prove that the sum of three consecutive cubes are divisible by $9$. Base case: Let $n=0$. Then $0^3 + 1^3 + 2^3 \equiv ...
0
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1answer
33 views

$\phi:A\to B$, $F:C^B \to C^A$. $F(f)=f\circ\phi$. prove that if $\phi$ is injective, then F is surjective [duplicate]

prove: let there be $\phi:A\to B$, $F:C^B \to C^A$. $F(f)=f\circ\phi$. prove that if $\phi$ is injective, then F is surjective. I did something but from some reason I haven't used the $\phi$ so ...
2
votes
2answers
88 views

prove\disprove if $f\circ g$ is invertible then $g\circ f$ is invertible

The question is to prove\disprove that if $f\circ g$ is invertible then $g\circ f$ is invertible. $f:A\to B$, $g:B\to A$. (f,g are functions) I tried to prove it but always got stuck, so I began ...
6
votes
2answers
51 views

prove/disprove that if $f\circ g$ injective and g is surjective, then f is injective

Question would be: prove/disprove that if $f\circ g$ injective and g is surjective, then f is injective. after thinking, I came to the conclusion that it's a proof. tried to prove it but it looks not ...
0
votes
1answer
28 views

What is the closed form of this series: $\sum_{n\geq 1}\frac{n^k{(-1)}^{n+1}}{n!}$ for $k<-10$ and for $k>1$?

I would like to check the closed form of this sum $$\sum_{n\geq 1}\frac{n^k{(-1)}^{n+1}}{n!}$$ , for an integer $k>1$ and $k<-10$. Note : I run some computation in wolfram alpha i have got ...
1
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0answers
25 views

A characterization of normal schemes (clarification of a statement of proposition)

The following is taken from 4.1 in Liu's book. Definition: A scheme $X$ is normal at $x \in X$ if $O_{X,x}$ is normal. $X$ is normal if it is irreducible and normal at every point. ...
3
votes
2answers
109 views

Real Analysis: Measure Zero

Show that the set $R^n$ x 0 has measure zero in $R^{n+1}$ This question has been asked before, I'm sure all the answers given are great but due to my relative novelty to real analysis I was unable to ...
0
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1answer
42 views

Open sets and measure zero

Show that no non-trivial open set in $R^n$ can have measure zero in $R^n$. Attempt at the solution: I am having a lot of difficulty attempting this question, I have read a lot of material on measure ...
3
votes
1answer
33 views

Help with a simple proof in predicate logic (with identity)

it's been a while since I've done formal logic, and I was trying to help a friend with a proof. His logic course is using Lemmon's Beginning Logic. Here's what he's supposed to show, without using ...
0
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0answers
61 views

Riemann Integration and measure zero

Show that if $A$ has measure zero in $R^n$, the sets $\overline{A}$ and $\mbox{Bd}(A)$ need not have measure zero. Attempt at a solution: I know we could use a counter example but I'm trying to ...
1
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5answers
72 views

Proof that $2^{2n}-1$ is not prime for $n \in \mathbb{N}, n > 1$

I notice that the number seems to be a multiple of 3: for n=2: $2^4 -1 = 15 $ for n=3: $2^6 -1 = 63$ for n=4: $2^8 -1 = 255$ How do I generalise?
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0answers
24 views

Proving concavity of a function

Let $f:[a,b]\rightarrow\mathbb{R}$ be twice-differentiable. Show that $f$ is concave if and only if $f''(x)\leq0$ for all $x\in[a,b]$. Moreover, if $f''<0$ for all $x\in[a,b]$, $f$ is strictly ...
0
votes
1answer
53 views

Need help understanding a proof about permutations in Rotman's ''Advanced Modern Algebra''

I need a hand in understanding the proof from "Advanced Modern Algebra" by Joseph J. Rotman of the following theorem: Let $\alpha \in S_n$ and let $\alpha = \beta_1...\beta_t$ be a complete ...
0
votes
1answer
36 views

Riemann Integration, question from Munkres

Let $[0,1]^2 = [0,1] \times [0,1]$. Let $f: [0,1]^2\rightarrow$ $\mathbb R$ be defined by setting $f(x,y) = 0$ if $x \not= y$ and $f(x,y) = 1$ if $x = y$. Show that $f$ is integrable over $[0,1]^2$ ...
0
votes
1answer
17 views

Well-formed formula a theorem in first-order system K

I'm trying to show that the following well formed formula is a theorem of K: $(\forall{x_i})\sim(A \to B) \to ((\forall{x_i})A \to (\forall{x_i})\sim B)$ Before going any further, I've rewritten it: ...
2
votes
2answers
42 views

Proof containing pairwise disjoint sets

I came across the following question while studying. Let $A,B,C,D$ be pairwise disjoint sets. Prove that if $|A| = |B|$ and $|C| = |D|$ then $|A \cup C| = |B \cup D|$. I thought of the fact that ...
8
votes
2answers
275 views

Axiom of Choice needed to “categorify” the cardinals?

I was playing around in $\mathsf{Set},$ trying to reduce it modulo isomorphisms to make a category $\mathsf{Card},$ letting the objects of $\mathsf{Card}$ be the isomorphism classes of $\mathsf{Set}$ ...
3
votes
1answer
60 views

Reading proofs vs. Attempting proofs, which one is more helpful?

In my discrete math course, I am often finding myself spending way too much time attempting a single math proof. I am starting to think that reading as many proof solutions as possible is a better ...
4
votes
1answer
35 views

Number of solutions to $|ax - bx| = a \;\text{or}\; b$?

While watching basketball tonight, I noticed that for 3, 4, and 6, $(6 \times 3) - (4 \times 3) = 18 - 12 = 6$. I thought this was a cool relationship and it led me to the following question: For ...
1
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4answers
71 views

If $0\lt y \le 1$, prove that there exists a unique positive real number $x$ such that $x^2=y$

I'm stumped. I don't want an entire solution, just a hint. If $0\lt y \le 1$, prove that there exists a unique positive real number $x$ such that $x^2=y$ The section in the book I'm on is the least ...
4
votes
2answers
102 views

Let $X$ an infinite $T_1$ space, then exist some subspace homeomorphic to $(\Bbb N,\tau)$ where $\tau$ is discrete or cofinite

My attempt to prove the statement of the title: if $X$ is infinite and $T_1$ with topology $\tau_1$ then any basis for $X$ is infinite too. If $X$ is $T_1$ then for any $x, y\in X$ with $x\ne y$ ...
1
vote
1answer
52 views

Proving that there exists a prime, p, between n! and n!+n iff p=n!+1

I have a conjecture (I believe this is not sophisticated enough to be a theorem) that there exists a prime, p, between n! and (n!+n) iff p=n!+1. I've tried a direct proof and a proof by contradiction ...
0
votes
1answer
37 views

Proving Inequality Statements From Proofs By Induction

Is it possible to prove $$1 + \frac{k}{2} + \frac{1}{2^{k+1}} \ge 1+ \frac{k+1}{2}\ \text{?}$$
0
votes
1answer
90 views

Proof of Euclid's Lemma in N that does not use GCD

I am looking for a proof of Euclid's Lemma, i.e if a prime number divides a product of two numbers then it must at least divide one of them. I am coding this proof in Coq, and i'm doing it over ...
3
votes
2answers
79 views

What is the function satisfying $\int_a^b f(x)\,dx =\int_a^b f'(x)\,dx$?

Let $$f:\mathbb R\rightarrow (0,\infty)$$ be a twice differentiable function such that $f(0)=1$, and further, for all $a, b \in \mathbb{R}$ such that $a < b$, $$\int_a^b f(x) \, dx =\int_a^b ...
2
votes
2answers
43 views

Whether a continuous function has fixed point or not when the domain and range are not $[0,1]$

Which of the following is false $?$ $A.$ Any continuous function from $[0,1]$ to $[0,1]$ has a fixed point. $B.$ Any homeomorphism from $[0,1)$ to $[0,1)$ has a fixed point. $C.$ Any bounded ...
2
votes
2answers
46 views

Delta epsilon argument in general

When I want to prove something in mathematics fe an expression goes to zero, I can either use basic rules of 'limits' or I can use the epsilon-delta method. I have a feeling that it's more consistent ...
1
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3answers
35 views

Need some hint about proving that if $X\cong Y$ and $X$ is $T_0$ then $Y$ is $T_0$

Im following the book Topology without tears of Morris (my first serious attempt to understand this area of modern maths as an amateur) and Im having a lot of trouble with proofs (proofs are a very ...
3
votes
1answer
46 views

$ \max \{ \underset{x \in X}{\sup } f(x)\ , \underset{x \in X}{\sup } g(x) \} = \underset{x \in X}{\sup } \{ \max \{ f(x), g(x) \} \}$?

Let $f,g: X \rightarrow [0, \infty )$ be bounded functions. Does hold that $$ \max \{ \underset{x \in X}{\sup } f(x)\ , \underset{x \in X}{\sup } g(x) \} = \underset{x \in X}{\sup } \{ \max ...
0
votes
2answers
33 views

Prove that all integers $n$ with $n \ge 12 $ is in the set.

$T = \{3k+7m: k,m \in \mathbb{Z}$, and $ k \ge 0, m\ge 0\}$ I assume I would do this by induction, so I want to show $3k+7m=n$ for $n\ge 12$. for the base case $n=12$. $3k + 7m = 12$ is true for ...
3
votes
3answers
134 views

prove if the function is vector space

Let $\Bbb R^3$ be 3-dimensional euclidean space. Is the set of all vectors in $\Bbb R^3$ that have the form $v=(v_1,v_2,v_3)$ where $5v_1-3v_2+2v_3=0$ itself a vector space? so my take: $v+w = ...
0
votes
0answers
32 views

Prove $n*2^n \leq 3^n$ using induction [duplicate]

I am trying to prove $n*2^n \leq 3^n$ for all $n \geq 1$ using induction. I tried to get it into the form $(n+1)*2^{n+1} \leq 3^{n+1}$ as follows: $n*2^n \leq 3^n$ $2n*2^n \leq 2*3^n$ $n*2^{n+1} ...
3
votes
3answers
69 views

Proving an inequality using mathematical induction

Using induciton, I have to prove following inequality: $$ 3^n > n2^n $$ I proved it for $n = 0$. Then assuming that the above is true, I try to prove it for $n+1$. So I start with: $$ (n+1)2^{n+1} ...
0
votes
0answers
57 views

Cyclic Group permutations

I am working on the following exercise question: Consider the following construction of a “keyed” hash function from Katz & Lindell (ex. 7.22 (1st ed.)/ 8.21(2nd ed.)). Gen : On input 1n , ...
0
votes
1answer
42 views

inclusion-exclusion principle proof (Without summations)

Suppose $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ are sets with $\mathcal{A} \cap \mathcal{B} \cap \mathcal{C} = \emptyset$. Then $| \mathcal{A} \cup \mathcal{B} \cup \mathcal{C} |$ = ...
0
votes
2answers
35 views

Proofs with for all statements including uniqueness and divides

Let $\mathcal{A}$ be a nonempty finite set of positive integers, with $\forall$ r $\in$ $\mathcal{A}$, $\forall$ s $\in$ $\mathcal{A}$ : r|s or s|r. (i). Prove $\exists$t $\in$ $\mathcal{A}$: t|a, ...
1
vote
2answers
64 views

About $−\vert x \vert\le x \le \vert x \vert$ in absolute values

So, i'm really strugling with this one, when studying the triangle inequality, the inequality $−|x|≤x≤|x|$ pops really often, however I just can't get why this is true. I've seen only two different ...
3
votes
0answers
50 views

A different way to prove homeomorphism between rectangles and discs under standard $\Bbb R^2$ topology

I was reading about the classical problem to prove homeomorphism between subspaces of $\Bbb R^2$ a rectangle of the kind $R=\{(x,y):|x|\le a \land |y|\le b\}$ and some disc $D=\{(x,y): x^2 + y^2 \le ...
1
vote
1answer
32 views

Banach Space Closed Subspace

Let $ \mathcal B$ be a Banach Space. Fix $z \in \mathcal B$ with $z \neq 0$. Consider the set $$A :=\{y-z : y \notin \operatorname{span} \{z\}, y \in \mathcal B\}.$$ Is it true that $\alpha z \notin ...
0
votes
1answer
28 views

Prove: $\exists !$ $t \in \mathbb{N}$ s.t. $\forall s \in \mathbb{N}$, $(t-9)s = 0$

I have a basic uniqueness proof to help me work on form: It should be obvious by simple inspection that the statement is true for t=9 and only for t=9. So my proof was this: Let $t=9$ then $9-9=0$, ...
1
vote
1answer
32 views

Proof of a six digit number [duplicate]

How can we prove that the number $142857$ is the only six digit number with the property that if I put the last digit before the first digit we get $5$ times our number so: ...
0
votes
1answer
36 views

Let $S$ be any set of statements. How do I concisely show that $\sim$ is reflexive, symmetric, and transitive on $S$?

The following problem is exercise 2.5.2 from "Mathematical Logic" by Ian Chiswell and Wilfrid Hodges (2007). I feel that the part about symmetry and transivity is a bit verbose and somewhat clumsy. ...
4
votes
1answer
123 views

Would like a hint for proving $(\forall x P(x)) \to A \Rightarrow \exists x( P(x) \to A)$ in graphical proof exercise on The Incredible Proof Machine

Update: Updated the title now that I've observed that we can use math in the title. I've also gone thru and removed dots. The tool expresses quantification using dots like this $\forall x.P(x)$ rather ...
0
votes
2answers
29 views

Find a family of open sets whose intersection is compact.

Does such intersection exists? im thinking about $An=(3+1/n;4+1/n)$ since $\bigcap An = [3,4] $ so its closed and bounded then its compact. Can someone please say whether its correct or not?
1
vote
0answers
30 views

Positivity of this improper integral

I am not relieved of the following problem and trip. Let $\varepsilon>0$ be a small parameter, $a>0$ be a given constant, $x_{\varepsilon}\in(0,a)$ be a given sequence such that ...