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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0answers
37 views

Determine set of t-norms verifying a property

As recalled in Scholarpedia, a t-norm $\top(a,b)$ is a function from $\mathbb{R}^{2}$ to $[0,1]$ verifying the following properties: Commutativity: $\top(a,b)=\top(b,a)$ Associativity: ...
1
vote
4answers
104 views

Proof that $f(x)=\frac{(\sin x)^3}{x}$ gets maximum in $(0,\infty)$.

I have this problem, and I got stuck in my proof Prove $f(x)=\dfrac{(\sin x)^3}{x}$ gets maximum in $(0,\infty)$. My Proof $$(1)\lim_{x \to 0+} \frac{(\sin x)^3}{x}= 0$$ $$(2)\lim_{x \to \infty} ...
1
vote
1answer
70 views

Let $f:[a,b]\to\Bbb R$ be as follows: $f(a)=A; f(b)=B$ and $f(x)=C$ for $a<x<b$. Show $f$ is integrable and the integral is $ C(b-a)$

Consider for real $a<b$ and real $A,B,C$, the function $f:[a,b] \to \mathbb R$ defined by $$f(x) = \begin{cases} A & x = a \\ B & x=b \\ C & a < x < b \end{cases}$$ I want to ...
26
votes
4answers
4k views

Why do proof authors use natural language sentences to write proofs?

I haven't read very many proofs. The majority of the ones that I've read, I've read in my first-year proofs textbook. Nevertheless, its first chapter expatiates on the proper use of English in ...
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0answers
52 views

Proving symmetry of metric (single linkage between clusters using arbitrary dissimilarity measure)

I am told to assume that our dissimilarity measure $d$ satisfies the properties required of it, what seems to be the definition of a metric: $d(x,y) \geq0 $ and $d(x,y)=0 \Longleftrightarrow x=y$ ...
3
votes
2answers
48 views

Prove that $4^{(p-1)/2}\equiv 1\pmod p$

If $p$ is a prime of the form $4k+3$, prove that $4^{\frac{p-1}{2}}\equiv 1\pmod p$. I was solving a problem and it came down to this. I have no idea how to prove it, I have tried. Any help would be ...
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0answers
30 views

Show f is integrable and integral is C(b-a)

Let $f:[a,b]\to\Bbb R$ be as follows: $f(a)=A; f(b)=B$ and $f(x)=C$ for $a<x<b$. Show $f$ is integrable and the integral is $ C(b-a)$ Consider for real $a<b$ and real $A,B,C$, the function ...
1
vote
1answer
25 views

Define $\phi:G/H \rightarrow G/K$ by $\phi(Ha)=Ka$. Prove: $\phi$ is a well defined function.

Let $H$ and $K$ be normal subgroups of a group $G$, with $H \subseteq K$. Define $\phi:G/H \rightarrow G/K$ by $\phi(Ha)=Ka$. Prove: $\phi$ is a well defined function. [That is, if $Ha=Hb$, ...
0
votes
1answer
27 views

How to proove that smallest upper bound exists und it is cleary determined?

Let X be a set. Then a relation '$\le$' on $\mathcal P(X)$ is defined by: $A \le B :\Leftrightarrow A \subset B$ . Let $\mathcal A \subset \mathcal P(X)$. One set $B \in \mathcal P(X)$ for which ...
0
votes
1answer
16 views

Lining a rectangular building square panels

I've been working on this for what feels like a lifetime now and I'm just not getting anywhere with it. I'm wondering if someone would be able to explain how to solve it for me? There is a ...
0
votes
1answer
13 views

Understanding how to prove a bijection into three sets

I understand how to prove if there is a bijection from A onto B. However, say that there is a bijection from A onto B and a bijection from B onto C. How would I prove that that there is a bijection ...
2
votes
2answers
21 views

Help proving this recurrence relation?

Let $P_n$ be the number of strings of length n formed from letters A, B, C, E, O, that do not contain two consecutive consonants (that is, B or C). For example, AABOCA and BACOOEBO satisfy this ...
0
votes
1answer
33 views

Question about a proof concerning abelian p-groups

I want to prove (convincing myself), why is this rght. In the proof of the lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for ...
1
vote
0answers
35 views

Verify why a thing of if this proof is right

I want to prove (convincing myself), why is this rght. In the proof of the lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some ...
4
votes
0answers
50 views

Prove this congruence

Let $p$ be a prime of the form $4k+3$ and $m$ an even positive integer less than $p-1$. Prove that $$1^m+2^m+\cdots+\left(\frac{p-1}{2}\right)^m \equiv ...
1
vote
1answer
30 views

Prove that it is transitive

Below is what I have so far. I'm pretty sure that it is transitive, but I'm not sure how to prove that it is. Prove that A is or isn't transitive.
2
votes
2answers
78 views

A doubt with a part of a certain proof.

Well, in the proof of the following lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at ...
2
votes
3answers
45 views

How can I prove that the square of an even number ends in 0/4/6?

I am trying to prove that the last digit of the square of an even number is either 0, 4, or 6 but I'm completely lost and have no idea how to tackle this problem.
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5answers
82 views

Series and sequences convergence with a certain condition.

Let $\sum_{n=1}^\infty (a_{n})$ converge. Let $\{n_{k}\}$ be a subsequence of the sequence of positive integers. For each $k$ define $b_{k}=a_{n_{k-1}+1}+...+a_{n_{k}}$ where $n_{0}=0$. Prove that ...
0
votes
1answer
37 views

How to prove distributive property of a determinant?

How to prove that $|A\cdot B| = |A|\cdot|B|$ where A and B are square matrices of the same size? P.S.: This proof is not mentioned in my textbook, nor was I able to find it on the web.
2
votes
1answer
34 views

Is there any good reason against referring to employed equations over the relation sign when establishing a new relation?

I need to write down a complicated proof for a paper, for which I need to employ equations that I established earlier for almost every new relation I show. I would consider it best for the reader, if ...
0
votes
2answers
57 views

$x,y$ are real, $x<y+\varepsilon$ with $\varepsilon$>0. How to prove $x \le y$? [closed]

$x,y \in \mathbb R : x \lt y + \epsilon : \epsilon \gt 0$ Then prove $x \le y$ .
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0answers
38 views

Essential part to undestand a proof . [duplicate]

In the proof of the the lemma suppose $G$ is a finite abelian $p$-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at ...
3
votes
3answers
86 views

Could this linear algebra proof be done without computation?

From page 95 of Hoffman & Kunze's Linear algebra: Let $T$ be the linear operator on $\mathbb{R}^2$ defined by $T(x_1,x_2)=(-x_2,x_1)$ Prove that if $B$ is any ordered basis ...
-1
votes
1answer
63 views

Prove that product of two continuous functions is continuous [closed]

Let $h,g$ be continuous functions Prove: The function $G:\mathbb R\to \mathbb R$ defined by $G(x)=h(x)\cdot g(x)$ is continuous
0
votes
1answer
37 views

Why do a coset is isomorphic to a certain set.

I have encountered with the proof of the next lemma suppose G is a finite abelian p-group, and let $C$ a cyclic subgroup of maximal order, then $G=C\oplus H$ for some subgroup $H$ at ...
0
votes
1answer
27 views

Determining equivalence classes

I have done (a), pretty straight forward. I understand an equivalence class as all the elements in the domain that map to the same result in the co-domain. For example in (mod 3), [|0|] would be the ...
0
votes
1answer
29 views

How to prove that lim sup $a_{n} \leq b$

Assume that $(a_{n})$ is a bounded sequence, prove that lim sup $a_{n} \leq b$ iff, for every $\epsilon > 0$, there exists an $N \in \mathbb{N}$ so that $n \geq N$ implies $a_{n} \leq b + \epsilon$ ...
1
vote
1answer
20 views

Writing skills: Proof of the relation between $\epsilon - \delta$ and open sets continuity

In order to check my math writing skills, I worked on writing the following basic proof. Theorem: If a function $f: X \to Y$ is continuous, then $G \subseteq Y$ is open implies that $f^{-1} (G)$ is ...
0
votes
0answers
49 views

How to show this is a graded ideal?

I have a very simple question, but since this is the first time I'm dealing with graded ideals and so on it seems more difficult than it really is. Suppose $V$ is a finite dimensional vector space ...
3
votes
2answers
48 views

Prove that the sequence $a_{n+1} = 2a_{n} - (a_{n})^2$ is bounded.

Prove that the sequence $a_{0} = \frac{1}{2}, a_{n+1} = 2a_{n} - (a_{n})^2$ is bounded. Assume that $0 < a_{n} < 1$ for every $n$ and $a_{0} = \frac{1}{2}$. Prof. used induction to prove that ...
0
votes
2answers
33 views

Proving Primness in a summation

I've been hitting my head against the wall for a little bit trying to figure out where to get started on proving (or disproving) the following: $\exists k \in \mathbb{Z} $ such that$ ...
3
votes
1answer
29 views

Is this exercise right, or something is wrong or missing.

I have to find the following limit For each positive integer $n$ define: $$a_n = \frac{1}{n}\left[\left(\frac{1}{n}\right)^2 + \left(\frac{2}{n}\right)^2 + ... + \left(\frac{n}{n}\right)^2 ...
0
votes
1answer
44 views

Verify if the following prove is right.

I need to prove the following: Suppose $f: [a,b] \to R$ is continous and $g: [c,d] \to [a,b]$ is differentiable. Define $F(x)= \int_{a}^{g(x)}f(t)dt$ for some $x \in [c,d]$. Prove that $F$ is ...
3
votes
2answers
91 views

Is $x = 2$ is the only real solution for $a^x + b^x = c^x$ when $(a,b,c)$ is a pythagorean triplet?

Take any pythagorean triplet $(a,b,c)$, we know, by the definition that: $$a^2 + b^2 = c^2$$ But take $$a^x + b^x = c^x$$ Is $x=2$ the only possible solution $\in \Bbb R$ in this case? How can this ...
1
vote
2answers
17 views

Closed set contains the set and its closure proof check

The problem is as following: $A\subset X$. Show that IF $C$ is closed set of $X$ and $C$ contains $A$, then $C$ contains the closure of $A$ Here is my proof, but I dont know whether I have the right ...
1
vote
0answers
31 views

Prove this limit for a general $f(x)$ and $g(x)$

$f(x)$ and $g(x)$ have the following property: for all $\epsilon > 0$ and all $x$, $$ \text{if} \space 0 < |x - 2| < \sin^2(\epsilon^2/9) + \epsilon \space \text{then} \space |f(x) - 2| < ...
2
votes
3answers
31 views

Proving $f(n + 1) > f(n)$ and is f injective?

If I have a function $f:\mathbb N \to \mathbb N$ defined for every $n \in \mathbb N$ by: $$f(n) = (n+1)!-1$$ How would I prove that $f(n+1) > f(n)$ for every $n\in\mathbb{N}$? Would it be ...
1
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3answers
39 views

Show that $A\cap B\subseteq A$ and $A\subseteq A\cup B$

$$A \cap B \subseteq A$$ My first step would be to write it as $(x \in A \land x \in B) \subseteq A$. Then I know by the following implication that is always true $P \land Q \implies P$. But I am ...
1
vote
3answers
72 views

If $f$ is continuous, $f(1) >1$ and $f(x+y)=f(x)f(y)$, then $f$ is increasing.

Consider the function $f$ with the following properties: $$\lim_{x\rightarrow 0} f(x) =1,$$ $$f(x+y)=f(x)\,f(y),$$ $$f(x) >0,\quad \forall x\in\mathbb{R},$$ $$ -\infty<x,y<\infty.$$ Show ...
1
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4answers
93 views

Proof: For every $x \in \mathbb R\,\exists n\in\mathbb Z,\,r\in[0,1):x=n+r.$

I still do not understand how to approach proofs. Any help would be appreciated. For every real number $x$, there exists an integer $n$ and a real number $r\in [0,1)$ such that $x = n + r$. ...
0
votes
1answer
56 views

Is $f : A \to P(A), a \mapsto \{a\}$ injective or surjective? [duplicate]

Given an arbitrary set $A$, let $f:A \to P(A)$ be the function defined for all $a \in A$ by "$f(a) = \{a\}$". How would you prove that $f$ is injective or surjective?
2
votes
1answer
37 views

Equivalence relation proof example. Starting off help

Now I need to prove its reflexive, symetric, and transitive! Now my biggest confusion is what do I let "a" equal? Obviously it will be an arbitrary element in N(sub 0). Any help would be great. ...
2
votes
1answer
45 views

having trouble saying this in a rigorous way

This is part of a problem I am working on and I have most of it figured out, but there is this one little piece that's kind of bothering me. I have a function $f:D \rightarrow R$ which is ...
0
votes
2answers
54 views

Prove the convergence of a sequence involving integrals

I need to prove the following: Assume $f:[a,b] \to R$ is continous, $f(x)\leq0$ for all $x \in [a,b]$, and $M=sup\{f(x):x \in [a,b]\}$.Show that: $$\{[\int [f(x)]^{n}dx]^{1/n}\} \to M$$ This result ...
1
vote
1answer
36 views

Proof involving Induction [duplicate]

Prove that for every integer n ≥ 1, we have $$ \sum_{i=1}^ni^3=\left(\frac{n(n+1)}2\right)^2 $$ Solve using Mathematical Induction, include the Inductive Step Base Case is that both the left and ...
31
votes
2answers
572 views

A strange integral

While browsing on Integral and Series, I found a strange integral posted by @sos440. His post doesn't have a response for more than a month, so I decide to post it here. I hope he doesn't mind because ...
0
votes
2answers
34 views

Proving that a certain series converge if and only if the a_n converges

I need to prove the following statement: Let $\{a_n\}$ be a sequence of real numbers.Prove that $\sum_{n=1}^\infty (a_{n}-a_{n+1})$ Converges iff $\{a_n\}$ converges. If $\sum_{n=1}^\infty ...
0
votes
1answer
28 views

“Natural” Homeomorphisms, Retracts and Knots

I have been trying to prove the following result for a few days now, and have made some amount of progress, but now I'm struggling. This is what I'd like to prove: Every proper knot, K has a retract ...