Questions tagged [examples-counterexamples]

To be used for questions whose central topic is a request for examples where a mathematical property holds, or counterexamples where it does not hold. This tag should be used in conjunction with another tag to clearly specify the subject.

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54 views

Counter example for Baire's Theorem

Theorem: Let $(X,d)$ be a complete metric space, and let $D_n, n\in \mathbb N$ be open, dense subsets of $X$. Then also $\bigcap_{n\in\mathbb N} D_n$ is dense in $X$. This statement is false if $X$ ...
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1answer
59 views

Hard inequality :$\sum_{cyc}\frac{a}{\sqrt[3]{a+b}}\leq a+b^{\frac{2}{3}}+c$

I'm interested by the following problem : Let $a,b,c$ be positive real numbers such that $a+b+c=1$ and $a\geq b \geq c$ then we have : $$\sum_{cyc}\frac{a}{\sqrt[3]{a+b}}\leq 1-b+b^{\frac{2}{3}}=...
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61 views

Counterexample to $ \frac{\mathrm{d}{y}}{\mathrm{d}{x}} = \frac{1}{\left( \frac{\mathrm{d}{x}}{\mathrm{d}{y}} \right)} $

I saw this question: Is $ \frac{\mathrm{d}{x}}{\mathrm{d}{y}} = \frac{1}{\left( \frac{\mathrm{d}{y}}{\mathrm{d}{x}} \right)} $? If this is true then see the following example: $y = sin(x)$, then $\...
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1answer
62 views

find 2×2 matrix such that $ A^2 =-I$

find 2×2 matrix such that $ A^2 =-I$ . I think that eigenvalue of $A^2$ is 0 or positive but eigenvalue of $-I$ is negative so we haven't matrix wih this condition.
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1answer
57 views

Set theory based on different logic

In HoTT book it is written that, We note that a set-theoretic foundation has two “layers”: the deductive system of first-order logic, and, formulated inside this system, the axioms of a particular ...
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1answer
62 views

If $f$ is absolutely continuous $\sqrt(f)$ may not be

The problem reads: Prove that if $ f:[0,1]\rightarrow(0,\infty) $ is absolutely continuous $ \sqrt{f} $ may not be. I am having trouble figuring out how to show this. I found that $x^2\sin\left(\...
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2answers
23 views

Questioning dense subset completeness (counterexample)

Let $X$ be a separable metric space and $A \subset$ X be countable and dense. Characterize the statements below as true or false (and why). If every Cauchy sequence in $A$ converges in $X$, $...
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1answer
39 views

Connection between composition and “inner composition” in closed monoidal categories

Let ${\mathcal C}$ be a symmetric closed monoidal category, $I$ its unit object, $\lambda_X:I\otimes X\to X$ the left unit morphism, and let me denote the internal hom-functor by a fraction $$ (X,Y)\...
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1answer
29 views

Uniformly Integrability and Non-Tightness

I want to costruct a measure space $(X,\mathcal{F},\mu)$ and a $\mathcal{C}\subset\mathrm{m}\mathcal{F}$, where $\mathrm{m}\mathcal{F}$ be the set of extended real-valued measurable functions on $X$, ...
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1answer
27 views

Find a group epimorphism $\mathbb{Z}_m \rightarrow \mathbb{Z}_{(m,n)}$ with kernel $n\mathbb{Z}_m$

Let $m$ and $n$ be positive integers and $\otimes=\otimes_\mathbb{Z}$. Denote the GCD of $m$ and $n$ by $(m, n)$. I proved $\mathbb{Z}_m \otimes \mathbb{Z}_n \cong \mathbb{Z}_{(m,n)}$ by considering ...
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1answer
67 views

Center of a group with order $p^aq^b$

Are there any examples of groups, $G$, such that $|G|=p^aq^b$ (where $p$ and $q$ are distinct primes and $a,b\geq 1$) and $|\mathrm{Z}(G)|=p^a$? ($\mathrm{Z}(G)$ denotes the center of $G$) I ...
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1answer
106 views

Nice olympiad inequality

Let's go for an olympiad inequality : let $a,b,c>0$ then we have : $$\sum_{cyc}\frac{ab}{a+b}\geq \frac{3\sqrt{3}}{2}\sqrt{\frac{abc}{a+b+c}}$$ My proof : $$\sum_{cyc}\frac{ab}{a+b}=\frac{(a^...
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1answer
37 views

Does a countable integral domain have only finitely many maximal ideals?

Does a countable integral domain have only finitely many maximal ideals? I've been thinking about this for awhile, I'd really appreciate a proof or counter example!! Thanks!
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3answers
48 views

Counterexample of Second fundamental theorem of Calculus if f is not continuous

Define $F(x)=\int_{a}^{x} f(t)\,dt$ on $[a,b]$, then by fundamental theorem of calculus, we know that if $f(x)$ is continuous then $F'(x)=f(x)$. Say we remove the condition that $f(x)$ is continuous ...
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69 views

Nice refinement of an inequality by Michael Rozenberg

It's related to this If $a+b+c=abc$ then $\sum\limits_{cyc}\frac{1}{7a+b}\leq\frac{\sqrt3}{8}$: In fact we have this refinement (wich I think much easier) : Let $a,b,c>0$ then we have : $$\...
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3answers
52 views

Construct $X$ s.t. it has a dense subset but elements in the complement of the subset have no sequence in the subset converging to them

Construct a topological space $X$ with the following property: There is a proper subset $Y$ of $X$ such that the closure of $Y$ is $X$ but if $c \in X-Y$ then there is no sequence in $Y$ converging to ...
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1answer
82 views

Undecidable cardinality

Let $S$ be an infinite subset of $\mathbb{R}$ with the property that the existence of $S$ can be proved within ZFC (and in particular the definition of $S$ does not invoke the negation of the ...
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2answers
41 views

Example of an experiment in which A, B, C are independent, but not pairwise independent

Can somebody give an example of process in which we have at least three events A, B, C and: P(A ∩ B ∩ C) = P(A) * P(B) * P(C) But A, B, C are not pairwise independent
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37 views

Example of non regular surface

I was reading definition of surface in differential geometry book which defined as follows A subset $S\subset \mathbb R^3$ is regular surface if $\forall p\in S $ there is open set in S such that ...
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1answer
51 views

$\{f_n \}$ is equicontinuous and pointwise converge to f is f uniformly continuous?

Let $\{f_n \}$ is equicontinuous and pointwise converge to $f$ now is $f$ uniformly continuous ? I can prove f is continuous but is this uniformly continuous? and we know every $f_n$ is uniformly ...
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1answer
32 views

Any algebraic function that can be expanded as infinite series with unbounded positive integral coefficients?

Cound one give examples of irrational algebraic function that can be expanded as infinite series with unbounded positive integral coefficients, like: $$f(x)=\sum_0^{\infty}a_i x^i,\qquad a_i\in \...
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1answer
25 views

Examples of commutative semirings satisfying $kb = hb + r \ \; \text{ iff } \; k = h \text{ and } r = 0$.

Let $\Bbb M$ be a commutative semiring. Setting $b = 1 + 1$, $\, \Bbb M$ also satisfies P-1: For every $k,h \in \Bbb M$ and $r \in \{0,1\}$ $$\tag 1 kb = hb + r \ \; \text{ iff } \; k = h \text{ and ...
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2answers
69 views

Easy counterexamples for each containment $Field\subsetneq ED \subsetneq PID \subsetneq UFD \subsetneq GCD \subsetneq Integral\space Domain$

$\mathbb Z$ could be used for an example of an Euclidean Domain that is not a Field. Can you give other easy examples for the other proper inclusions?
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1answer
27 views

Does every linear injective open map between Banach spaces map closed sets to closed sets?

Let $X$ and $Y$ be Banach spaces and $T \in L(X,Y)$, i.e. $T$ is a linear continuous map from $X$ to $Y$. Further let $T$ be injective and open. Does $T$ map closed sets to closed sets? I know this ...
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1answer
25 views

What are some examples of statement-reason format for ratios of segments?

I'm wondering what some examples might be for a proof (statement/reason format) that required you to find the length of two segments if you know that their ratio is $x:y$ and that the total length is $...
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1answer
42 views

Convergence to standard normal distribution but law of large numbers does not hold. Difficult example

Let $X_1, X_2,...$ be a sequence of random variables (not necessarily independent or identically distributed) Give an example of a sequence such that $\sum_{i=1}^n ({X_i-\mu}) \over \sqrt{n}$ ...
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1answer
68 views

Is the unit object in a closed monoidal category always integral?

I.Bucur and A.Deleanu in their "Introduction to the theory of categories and functors" define integral object in a category ${\mathcal C}$ as an arbitrary object $I$ that satisfies the following two ...
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162 views

Topological spaces which are not pseudometrizable.

Let $(X,\tau)$ be a topological space. Then we know some conditions under which $(X,\tau)$ is metrizable (see for example this and this). It is also clear from these theorems that not every ...
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18 views

If $f = o(g)$ and $h = o(p)$, what can we say about the asymptotic behavior of $f + g$?

I would very much like to say that $f + h = o(\text{whichever is bigger between g and p})$ but am a little bit worried if we have $g$ and $p$ oscillating as (say) $x \to \infty$. If we add the ...
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1answer
30 views

Find two functions $f,g$ such that $\displaystyle\lim_{x\to a}f(x)=A$ and $\displaystyle\lim_{y\to A}g(y)=B$

Find two functions $f,g$ such that $\displaystyle\lim_{x\to a}f(x)=A$ and $\displaystyle\lim_{y\to A}g(y)=B$, but $\displaystyle\lim_{x\to a}g(f(x))\neq B$. I wrote the definitions for both limits. ...
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2answers
30 views

Operator with given spectrum which is not projector.

I'm stuck in making an example of such operator $A$, that spectrum of $A$ is $\{0,1\}$, but A is not a projector. Could you give me such example, please
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1answer
48 views

Is the convex hull of a countable set a Borel set?

The convex hull of a subset $X$ of $\mathbb{R}^2$ is the smallest convex subset of $\mathbb{R}^2$ containing $X$. My question is, if $X$ is countable, then is the convex hull of $X$ necessarily a ...
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12 views

Example of fixed point free of order 2 which is not inverse mapping

I just learn about fixed point free of order 2 and when G is finite, this is nothing but map $\phi (x)= x^{-1}$. I am looking for example of fixed point free of order 2 which is not the map which $\...
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0answers
24 views

Concrete example of Lie algebra $\mathfrak{g}$ with $x$ s.t. $x$ is in the radical of $\mathfrak{g}$ but not the radical of the Killing form?

Let $\mathfrak{g}$ be a Lie algebra over algebraically closed field $k$ of characteristic $0$. The radical $R(\mathfrak{g})$ is the largest solvable ideal of $\mathfrak{g}$. The Killing form $\kappa$...
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58 views

Is there is group which can not be made into ring with any operation? [duplicate]

I wanted to have example of group which can not be ring? I think if we have non abelian group with some operation we can not proceed to ring . Is it correct or it required to more argument ? ...
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1answer
62 views

Is the interior of a Jordan curve a Borel set?

A Jordan curve is a continuous closed curve in the plane with no self-intersections. My question is, is the interior of a Jordan curve always a Borel set? If not, is the interior of a convex Jordan ...
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4answers
58 views

Differentiability class: Example of maps that are $C^k$ but not $C^{k+1}$.

Is there a classical example for the fact that the differentiability class satify $$ C^{k+1} \subsetneq C^{k} $$ I'm interested in the $C^{k+1} \neq C^{k}$, then is I'm looking for a classical ...
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1answer
32 views

Quotient ring of the zero-divisor ideal is a flat module

Let $A$ be a commutative unitary ring. In the D. G. Northcot's Multilinear Algebra it is claimed that if a proper ideal $I$ of $A$ contains a zero non-divisor element, say $a$, then the $A$-module $\...
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0answers
16 views

Non Markov chain process.

Consider $\xi_n $ are independent, non-negative, equal distributed random variables. Let $S_n = \xi_1 + \dots + \xi_n$. Let $N_t = \max \{n : S_n \le t\}$. Does there example of such process, which ...
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2answers
55 views

Properties of a derivative on a compact interval

Suppose a function $F$ is differentiable on an interval $(a,b) \supset [0,1]$. Denote its derivative by $f$, and suppose that $f > 0$ on $[0,1]$. Question 1: Is it true that $f$ can be bounded ...
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39 views

Find sequences such that …

Let $c\in \mathbb{R}$. Find two sequences $(a_n)_n$, $(b_n)_n \subset \mathbb{R}$ with: (i) $\lim\limits_{n\to\infty}a_n=\infty, \lim\limits_{n\to\infty}b_n=0 $ and $\lim\limits_{n\to\infty}a_nb_n=c$ ...
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16 views

Graded endomorphisms factoring through nilpotent ones

Let $M$ be a finitely generated 2-graded $k[x,y]$-module, concentrated in nonnegative degrees, and $f$ be a degree $(n,m)$-endomorphism of $M$. We can consider $f$ as a morphism $M\langle n,m\rangle\...
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2answers
61 views

What is a good example of prefix induction?

the Wikipedia article for Mathematical induction introduces a few variations of the classic principle, such as the strong induction. The strong induction comes with a few examples, namely the closed ...
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50 views

Counterexample in topological vector space.

Let's $X$ be a topological vector space. So it has two operations $+$ and $\times$. As we know these operations could be continous. But can it be that one of this operation is continous, but the ...
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1answer
32 views

Basic theorem for solvable groups not true for nilpotent groups - counterexample.

it's my first question on MathStackExchange so please be tolerant. Let H be a normal subgroup of group G. If H and G/H are both solvable, then G is solvable. But H nilpotent and G/H nilpotent doesn't ...
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1answer
55 views

Example of noncompact space in which every real valued continuous function on it is uniformly continuous

I wanted to find Example of non-compact metric space $(X,d)$ such that every real-valued continuous function is uniformly continuous My attempt: $X$ is an infinite set $d$ is a discrete metric. ...
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0answers
60 views

Linear bounded operator from $L^p[0,1]$ to itself whose range consists of continuous functions.

Let $T\colon \mathbb L^p[0,1]\to \mathbb L^p[0,1]$, $1<p<+\infty$, be a linear bounded operator such that $\operatorname{Im}(T)$ is contained in the space of continuous functions. It was shown ...
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1answer
29 views

Cantor's Intersection Theorem iff complete metric spaces?

If a metric space is complete, we know that the Cantor's Intersection Theorem holds. Does the converse also hold? And if not, what is a suitable counterexample for the same? Also, if the converse ...
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1answer
37 views

Pointwise convergence vs convergence in measure [closed]

I would be very grateful if I could be given an example of a sequence convergent pointwise but it is not convergent in measure and an example of a sequence convergent in measure but it is not ...
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1answer
273 views

“Counterexample” for the Inverse function theorem

In my class we stated the theorem as follows: Let $\Omega\subseteq\mathbb{R}^n$ be an open set and $f:\Omega\to\mathbb{R}^n$ a $\mathscr{C}^1(\Omega)$ function. If $|J_f(a)|\ne0$ for some $a\in\Omega$...