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.

5
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0answers
49 views

Examples of surfaces with constant curvatures

I'm looking examples of surfaces in $\mathbb{R}^3$ such that the Gauss and mean curvatures are both constants. For this assumptions, I saw in the Montiel and Ros'book that only the planes, right ...
2
votes
3answers
343 views

Eigenvalue and similar matrices

if $A$ and $B$ are two $n\times n$ matrices with same eigenvalue having same algebraic and geometric multiplicity. Does $A$ and $B$ are similar? If $A$ is diagnalizable then the claim is true. But ...
1
vote
2answers
63 views

Suppose that $f: \mathbb R^n \to \mathbb R^n$ is a bijection and $n\geq2$. Can $f$ send every open set onto non-open set?

Suppose that $f: \mathbb R^n \to \mathbb R^n$ is a bijection and $n\geq2$. Can $f$ send every open set onto non-open set? I do not know what exactly to write about this problem. Did I try anything? ...
0
votes
0answers
111 views
+100

Counterexample to the primality test

This is a generalization of this claim . Can you provide a counterexample to the following claim? Let $n$ be a natural number greater than two . Let $r$ be the smallest odd prime number such that ...
1
vote
0answers
33 views

Can this set-theoretic vector space be endowed with this operation so that everything is fine?

Mike asked in this question to find a model of vector space with additional operation $\wedge$ so that we have $a \wedge (a+b) = (a+b) \wedge b = a \wedge b$ and $\wedge$ is non-distributive. I built ...
2
votes
1answer
65 views

Do these functions exist? [duplicate]

I created this question, but, I do not know the answer: Is there a function $f: \mathbb R \to \mathbb R$ such that for every interval $I \subseteq \mathbb R$ we have $f(I)=\mathbb R$? It seems to ...
1
vote
0answers
48 views

Compositeness test for numbers of the form $E_n(b)=\frac{b^{2^n}+1}{2}$

Can you provide a proof or a counterexample for the following claim? Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ . Let $E_n(b)= \frac{b^{2^n}+1}{2} $ where $b$ is an odd ...
0
votes
0answers
31 views

Defining Multiplication on Strongly Homogeneous Commutative Semigroups

Let us call any commutative semigroup $(S, +)$ strongly homogeneous if it satisfies the following three properties: P-1. Every endomorphism on $S$ is a bijection. P-2. Any two endomorphisms on $S$ ...
0
votes
0answers
23 views

Looking for an example of Lusin Theorem

Is there a particular short example and not hard to understand (could be basically trivial but not totally trivial) of the Lusin's Theorem that you can provide please? I have found a related question ...
1
vote
2answers
50 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 $X$ ...
0
votes
2answers
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 $\...
1
vote
1answer
58 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.
0
votes
1answer
47 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 ...
0
votes
1answer
60 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(\...
0
votes
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 $X$, $...
2
votes
1answer
33 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)\...
1
vote
1answer
27 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 $X$, ...
0
votes
1answer
26 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 ...
1
vote
1answer
65 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 ...
2
votes
1answer
101 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^...
0
votes
1answer
35 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!
2
votes
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 ...
-1
votes
1answer
62 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 : $$\...
0
votes
3answers
48 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 ...
1
vote
1answer
76 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 ...
0
votes
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
3
votes
1answer
34 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 ...
2
votes
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 ...
0
votes
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 \...
1
vote
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 ...
1
vote
1answer
53 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?
1
vote
1answer
24 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 ...
1
vote
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 $...
1
vote
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 $\sum_{i=1}^n ({X_i-\mu}) \over \sqrt{n}$ ...
1
vote
1answer
54 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 ...
5
votes
2answers
157 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 ...
0
votes
0answers
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 ...
1
vote
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. ...
0
votes
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
2
votes
0answers
34 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 ...
0
votes
0answers
11 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 $\...
1
vote
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 $\kappa$...
3
votes
0answers
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 ? ...
-1
votes
1answer
58 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 ...
1
vote
4answers
57 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 ...
0
votes
1answer
30 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 $\...
0
votes
0answers
15 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 ...
4
votes
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 ...
0
votes
2answers
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 $\lim\limits_{n\to\infty}a_nb_n=c$ ...
0
votes
0answers
15 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\...