Examples and counterexamples are great ways to learn about the intricacies of definitions in mathematics. Counterexamples are especially useful in topology and analysis where most things are fairly intuitive, but every now and then one may run into borderline cases where the naive intuition may ...

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4
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8answers
405 views

Does a non-abelian semigroup without identity exist?

I was introduced to semigroups today and had a question. So all the examples of semigroups I was given were either monoids or groups. So I was curious, does there exist a semi-group which is not ...
2
votes
6answers
131 views

What would be an example of a magma such that $x\cdot(x\cdot x)\neq (x\cdot x)\cdot x$?

Let $(M,\cdot)$ be a non-associative magma. What would be an example of $M$ such that there exists $x\in M$ such that $x\cdot(x\cdot x)\neq (x\cdot x)\cdot x$?
0
votes
0answers
8 views

Vanishing cech cohomology of a concratible space.

Let $C$ be a concratible space. Is it possible to find a sheaf $F$ such that two Cech cohomology groups don't vanish? E.g. Can I find a sheaf (or a local system) and two integers $i,j$ such that ...
19
votes
12answers
2k views

An easy example of a non-constructive proof without an obvious “fix”?

I wanted to give an easy example of a non-constructive proof, or, more precisely, of a proof which states that an object exists, but gives no obvious recipe to create/find it. Euclid's proof of the ...
1
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1answer
20 views

Relation between convergence in distribution and in probability

Does convergence in distribution imply convergence in probability ? I suppose no, but I need a counterexample. Does anyone know any counterexamples ?
3
votes
2answers
233 views

Example of non-Abelian group with 4, 5, or 6 elements of order 2

Is there any example for non-Abelian group in which has more than $3$ elements of order $ 2$, but has less than 7 elements of order $2$?
2
votes
3answers
173 views

Basically disconnected space which is not extremally disconnected

Space $X$ is basically disconnected if every cozero-set has an open closure. Every extremally disconnected space is basically disconnected But i think the converse fails. The one-point ...
33
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18answers
6k views

Examples of mathematical induction

What are the best examples of mathematical induction available at the secondary-school level---totally elementary---that do not involve expressions of the form $\bullet+\cdots\cdots\cdots+\bullet$ ...
13
votes
1answer
636 views

Difficulty in finding a counterexample

I am finding difficulties in finding a counterexample that if $f\colon (0,\infty) \to(0,\infty) $ is uniformly continuous, this implies that $$\lim_{x\to \infty} \frac{f(x+\frac{1}{x})}{f(x)} =1.$$
1
vote
1answer
23 views

Properties of Jacobson radical

I'm looking for an example of ring epimorphism $\varphi:R\rightarrow S$ such that the natural homomorphism $\tilde\varphi:J(R)\rightarrow J(S)$ is not surjective, where $J(R)$ is a Jacobson radical.
2
votes
2answers
81 views

An example of quasigroups with no identity

If you scroll to the bottom of this page, there is a table claiming quasigroups have divisibility but not identity (in general). What would be some examples of a quasigroup without an identity ...
1
vote
2answers
71 views

Can an algebraic structure such that $x+a=x+b$, have solutions for all $a,b∈K$ exist?

Does there exist an algebraic structure $(K,+)$ such that equations of the form $x+a=x+b$, $a\neq b$ have solutions for all $a,b\in K$?
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1answer
21 views

An example of covergence to an exponential distribution, the role of continuity

I got a probability problem I can solve, but my solution does not use an assumption which is given in the formulation of the problem. I am afraid that this is might be a sign that my solution is ...
1
vote
2answers
31 views

Does every ball of boundary point contain both interior and exterir points?

My question is If $x$ is a boundary point of $S$ ($S$ is subset of $R$), does every ball of $x$ contain both interior points and exterior points of $S$? I think this is false. Since $R$ is union of ...
0
votes
0answers
27 views

Nyquist–Shannon Sampling Theorem Counter Example?

I was learning about the Nyquist theorem regards signal processing the area of interest which I will rephrase below: Given a signal lasting infinitely long with a maximum frequency of f, then you can ...
27
votes
0answers
467 views

Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
3
votes
0answers
65 views
+100

mononotone and pseudomonotone operators in current research

I know that the following question is quite broad for this forum. But I am interested in references or any other ideas. Can anyone provide some examples of applications of monotone or pseudomonotone ...
3
votes
3answers
130 views

Need example for a topological space that isn't connected, but is compact

We know the topological space $(R,τ_1)$ is a connected space but it is not compact, $(R,τ_+)$ (which generated by $[a,b[$) is not connected space and it is not compact space, and $(R,τ_{cf})$ is ...
287
votes
32answers
18k views

Examples of apparent patterns that eventually fail

Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of 'proof'. I receive responses like: "surely if the Collatz ...
3
votes
4answers
78 views

A function not differentiable exactly two points of $[0,1]$. construction of such a function is possible?

Can a continuous function on $[0,1]$ be constructed which is not differentiable exactly at two points on $[0,1]$ ?
2
votes
1answer
53 views

Example: convergence in distributions

Give an example $X _n \rightarrow X$ in distribution, $Y _n \rightarrow Y$ in distribution, but $X_n + Y_n$ does not converge to $X+Y$ in distribution. I got a trivial one. $X_n$ is $\mathcal ...
2
votes
2answers
32 views

counter example of equicontinuous

Consider the functions on $[0,1]$: $f_n(x)=nx$, when $x$ is between $0$ and $1/n$ $f_n(x)=2-nx$, when $x$ is between $1/n$ and $2/n$ $f_n(x)=0$, otherwise How to see it is not (uniformly) ...
63
votes
9answers
2k views

Problems that become easier in a more general form.

When solving a problem, we often look at some special cases first, then try to work our way up to the general case. It would be interesting to see some counterexamples to this mental process, i.e. ...
7
votes
0answers
54 views

Example of continuous function whose Fourier series doesn't converge on an uncountable dense set.

According to a well-known theorem (Theorem 5.12 in Rudin's Real and Complex Analysis), there is a dense $G_\delta$ set of continuous periodic functions $f:\mathbb{R}\to\mathbb{C}$ such that the ...
1
vote
2answers
25 views

Example for a sequence of continuous functions which converge to a continuous function pointwise but not uniformly on a compact set.

My actual aim is to verify that each of the conditions in Dini's theorem is essential or not.Theorem says that A sequence {$f_{n}$} of continuous functions defined on a compact set $E$ converges to ...
1
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1answer
34 views

Example of Two-point Remainder that are not homeomorphic

We that any two compactification $c_1 N$ and $c_2 N$ of the space $N=D(\aleph_0$) that have finite remainders of the same cardinality are homeomorphic, and yes can be incomparable with respect to the ...
0
votes
3answers
64 views

Example for why L'Hospital's rule demands that the denominator is locally not $0$.

We're looking at the limit of a function $f(x):=\frac{g(x)}{h(x)}$ for $x \rightarrow x_0$. Since L'hospital's Rule demands that the denominator function $h(x) \neq 0$ for a neighbourhood of $x_0$, ...
2
votes
1answer
52 views

An example of $_AM$ is simple but $M^\star_A$ is not?

Let $A$ be a finite dimensional $k$ algebra, $k$ is a field. It is evident that the duality functor ($(-)^\star=hom_k(-,k)$) preserves the simplicity in the case of finitely generated $A$- modules ...
1
vote
1answer
31 views

Visualize and define a vector space without dot / inner product

I'm trying to rebase my know how in linear algebra, restart from scratch to get a more formal and useful set of definitions to help me with computer programming stuff . One of the first concepts is a ...
3
votes
2answers
158 views

Is the converse of Sylvester's inertia law true?

Let $A,B$ be $n\times n$ symmetric matrices. Assume the positive,negative,0 index of $A$ and $B$ are the same. (That is, they have the same inertia) Then, are $A,B$ congruent? What is a ...
8
votes
1answer
325 views

A counter example in obstruction theory

Let $K$ denote a simplicial complex and $Y$ some topological space. Let us also denote by $K^n$ the $n$-skeleton of $K$. I would like to have an example for the following situation: There is a map ...
6
votes
1answer
78 views

An example of a simple infinite $2$-group

Is there an example of a simple infinite $2$-group? Informations If a $2$-group is Artinian I know that it also locally finite, so the simple $2$-group cannot be Artinian. Take the subgroup ...
0
votes
2answers
68 views

Compactness: Opens vs. Neighborhoods

Disclaimer: This thread is a record of thoughts. Discussion Given a compact set. Do mere neighborhood covers admit finite subcovers? $$C\subseteq\bigcup_{i\in I}N_i\implies C\subseteq ...
2
votes
2answers
33 views

Groupoid with division

I'm searching an example of a grupoid with division which is not a quasigroup. A grupoid $(G, \cdot)$ is with division if $a\cdot G=G\cdot a=G$. I was thinking to try $(\mathbb{Q},\cdot)$, where ...
0
votes
1answer
26 views

Do there exist semi-local Noetherian rings with infinite Krull dimension?

Do there exist semi-local Noetherian rings with infinite Krull dimension? As far as I know, Nagata's counterexample to the finite dimensionality for general Noetherian rings is not semi-local.
1
vote
2answers
156 views

Are there algebraic subgroups of Lie groups which happen to be Lie groups on their own but not Lie subgroups?

Let $G$ and $H$ be Lie groups and $i:H\to G$ an injective group homomorphism, not necessarily continuous. Once $i$ is continuous, it follows easily that $i$ is differentiable and even an immersion. ...
0
votes
1answer
66 views

Non-subdifferentiable convex function on a normed space

Is there any convex function $f$ on a normed space $X$ such that $\partial f(x)=\emptyset$ at every $x\in X$?
0
votes
2answers
42 views

Linear algebra questions

$M$ and $N$ subspaces of $V$. Give examples of $M$ and $N$ such that $M\cup N$ and $M\setminus N$ are not subspaces.
0
votes
1answer
28 views

Proof by counter-example

As is probably obvious by the title I cannot work out any set of numbers which disporoves the following conjecture $$ a^2 > b^2 => a > b $$ where A and B are real numbers. Anyone to give a ...
2
votes
0answers
27 views

Is a limit of a sequence of distribution functions is necessarily a distribution function

I got a question: "is a limit of a sequence of distribution functions is necessarily a distribution function". My answer is NO. I have a counterexample: $F_n(x)=0$ if $x<n$ and $1$ otherwise. It ...
1
vote
1answer
32 views

Differentiable continuous function whose derivative is not continuous [duplicate]

Is there a function which is continuous and differentiable, but is not smooth function? By smooth I mean having continuous derivative. For example, the derivative of $f(x)=x|x|/2$ is $f'(x)=|x|$ ...
2
votes
1answer
99 views

Weaker definitions of Lie subgroups

A Lie group $H$ is called a Lie subgroup of a Lie Group $G$ if there is a map $i:H\to G$ which is (a) an injective immersion and (b) a group homomorphism. My questios are: What happens if we replace ...
2
votes
1answer
104 views

Counter-example to Cauchy-Peano-Arzela theorem

I was looking for a counter-example to Cauchy-Peano-Arzela theorem. I found this paper (in french) from Dieudonné. [acta.fyx.hu] Take $E = c_0$ to be the space of real sequences converging to $0$, ...
11
votes
6answers
463 views

Just How Strong is Associativity?

A friend of mine is using a lot of algebra that is not associative for an advanced Chemistry project. We were discussing it recently and I found it rather amusing how often she said things like ...
0
votes
1answer
27 views

Example of a non-abelain finite group $G$ with $G/N$ abelian and infinite group $G$ with $G/N$ finite

Have not been able to think of a examples with the following properties: Example of a non-abelian finite group $G$ with property that $G/N$ is abelian for every non-trivial normal subgroup $N$ of ...
2
votes
0answers
62 views

An example of wreath product

I was analyzing the following example of wreath product of groups. Let $\mathbb{Z}_2$ be the cyclic group of order two and $\mathbb{Z}$ be the usual additive group of integers. Consider the ...
0
votes
1answer
40 views

How to find example such $A^{(n)}$ are different each other

let denoted $A'$ Derived set,and define $$A'=A^{(1)},(A^{(1)})'=A^{(2)},\cdots,(A^{(n)})'=A^{(n+1)}$$ Question: Take example the set $A$,such $A^{(k)},k=1,2,\cdots,n+1$ are different ...
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vote
2answers
36 views

A question about the orders of the elements of a group [duplicate]

Let $m$ and $n$ be to positive integers strictly larger than $1$. Is it possible to find a group $G$ in which there are two elements, say $a$ and $b$, such that the order of $a$ is $m$, the order of ...
1
vote
1answer
23 views

Example of a unbounded projection

Let $H$ be a Hilbert space over $\mathbb{K}$. Let $T:H\rightarrow H$ be a linear transformation such that $T^2=T$. What is an example of $T$ such that $T$ is unbounded?
7
votes
4answers
627 views

Simple (even toy) examples for uses of Ordinals?

I want to describe Ordinals using as much low-level mathematics as possible, but I need examples in order to explain the general idea. I want to show how certain mathematical objects are constructed ...