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Questions tagged [examples-counterexamples]

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 disagree with the actual situation that follows from the definitions. This tag should be used only in conjunction with another tag to clearly specify the subject.

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Does a topology with countable elements imply that the topology is second countable?

Does a topology with underlying space having countable elements imply that the topology is second countable? Today, the above question come into my mind. This seems very intuitive to me, but I am not ...
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Examples about “union of two Lie algebra is not Lie algebra” [on hold]

Could you give a some examples about "union of two Lie algebra is not Lie algebra".
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Primality test for numbers of the form $N=k \cdot 3^n-1$

Can you provide proof or counterexample for the claim given below? Inspired by Lucas-Lehmer-Riesel primality test I have formulated the following claim: Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(...
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1answer
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Useful bijections

Could someone please provide me with some useful bijections one ought to know for an upcoming examination on cardinality with an emphasis on proofs? For example, the bijective mapping $f : (-1, 1) \...
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Minimal order of a counterexample to Wall’s conjecture

There used to be once a rather well known and interesting conjecture, that was formulated by Gordon E. Wall: The number of maximal subgroups of a finite group $G$ does not exceed $|G|$ That ...
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An example of an even numbers $n,k$ whose $gcd$ is three.

I am looking for even integers $n$ and $k$ such that $k$ does not divide $n$ and $\gcd(n,k) = 3$. Is this possible? With the help of some online tools I tried, but every time I am not getting the ...
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Examples of applying Dirichlet's approximation.

I've seen many examples of Dirichlet's approximation being proven , or other questions regarding to the theory of the approximation on this site and others but I would like to see a concrete example ...
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344 views

Semigroups with no morphisms between them

Given two monoids we always have a morphism from one to the other thanks to the presence of the identity element. Are there examples of non-empty semigroups that have no morphisms from one to the ...
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1answer
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Understanding HNN extensions: intuition, examples, exercises.

What is an HNN extension? What would be some elementary, intuitive examples of them and what exercises involving them would you suggest? The Wikipedia definition is easiest to get to, since neither ...
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1answer
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Counterexample: linearly ordered sets for which there exists more than one isomorphism

In my axiomatic set theory notes, there appears that, if $A$ and $B$ are well-ordered isomorphic sets, then there exists one isomorphism between them. However, as a side note, it is stated that this ...
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2answers
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Counter example for integral differing from limit of integrals

On exercise $4.U$ of Bartle's Element of Integration, we are asked to give a counter example on a sequence $f_n$ of functions such that $\lim_{n\to\infty}f_n=f$, $\int f =\lim_{n\to\infty}\int f_n$ ...
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What is a non-alternating simple group with big order, but relatively few conjugacy classes?

I'm not sure if this question is legal. I'm writing my BsC thesis on character theoretical calculations and I have already calculated a lot of character tables (a ...
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1answer
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Equal nr of $\mathscr D$-classes, different nr of idempotents

Are there examples of (finite) semigroups $S$ and $T$ such that they have the same 'number' of $\mathscr D$-classes, $S$ has idempotents and $T$ doesn't? Alternatively, they both may have idempotents, ...
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A Dedekind domain without prime elements

We know examples of non Noetherian Prüfer domains, which do not contain any irreducible elements. On the other hand, a Dedekind domain (not being a field) always contains irreducible elements since ...
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1answer
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What is the difference between optimization on Banach space versus optimization on Hilbert space?

In Chapter 4 of this book, it says, Suppose now that we are interested in the more general situation of optimization in some Banach space $B$. In other words the norm that we use to measure the ...
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Find $H<G$ so that $\{(x, y) | xx^{−1} y^{−1} \in H\}$ is not an equivalence relation on $G$.

The question is as follows: Find an example of a group $G$ with a subgroup $H$ so that $$\{(x, y) | xx^{−1} y^{−1} \in H\}$$ is not an equivalence relation on $G$. I've just been working on this ...
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2answers
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Basic question about convexity

A convex function is defined as one that satisfies the following condition for $p_1 + p_2 = 1$. $$f(p_1x_1 + p_2x_2) \leq p_1f(x_1) + p_2f(x_2),$$ Does this imply that for all $\lambda \leq 1$ $$...
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1answer
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Primality test for Mersenne numbers using the fourth Chebyshev polynomial of the first kind

Can you provide a proof or a counterexample for the claim given below? Inspired by Lucas-Lehmer test I have formulated the following claim : Let $T_n(x)$ be the nth Chebyshev polynomial of the ...
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2answers
308 views

Example of compact Riemannian manifold with only one closed geodesic.

The Lyusternik-Fet theorem states that every compact Riemannian manifold has at least one closed geodesic. Are there any easy-to-construct1 examples of compact Riemannian manifolds for which it is ...
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2answers
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Baire's Theorem: Examples for open dense subsets

Theorem (Baire): Let $(X,d)$ be a complete metric space and $(D_n)_{n \in \mathbb{N}}$ a family of open dense subsets of $X$. Then $\bigcap_{n \in \mathbb{N}} D_n$ is also dense in $X$. This is the ...
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1answer
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Almost Lindelöf spaces.

I'm working on the study of almost Lindelöf spaces and I'm stuck searching a counterexample. First, the definition. Let $X$ be a topological space. We say that $X$ is an almost Lindelöf space if ...
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Example of IVP $dx/dt=f(x,t)$, $x(0)=1$ with multiple bounded-time maximal solutions

Recently, I have been studying the existence and uniqueness of the solutions of IVP. I am wondering if anyone can give an example of the function $f(x,t)$ for the IVP $dx/dt=f(x,t), x(0)=1$ that ...
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1answer
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I need a $T_D$ topological space that is not sober

in the book Frames and Locales by Jorge Picardo are defined two types of spaces: Sober spaces where the only meet-irreducible open sets are those in the form $X\setminus x^-$, where $x^-$ is the ...
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1answer
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Easy to understand real world example for pde with only weak solutions

After taking a course of ODEs, I began reading about the theory of weak solutions. Without any examples the author claimed that i.e. the function being differentiable twice in the interior of the ...
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1answer
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Counterexamples for $\lim f(x,y)=\lim \left(y\cdot \frac{f(x,y)}{y}\right)$.

What are some examples of functions $f(x,y)$ satisfying the conditions below? $\displaystyle\lim_{(x,y)\to(0,0)} f(x,y)$ does not exist. $\displaystyle\lim_{(x,y)\to(0,0)} \frac{f(x,y)}{y}$ exists. ...
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1answer
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Uniform convergence does not guarantee convergence of integrals when the domain has infinite measure

Let $(X,\Sigma,\mu)$ be a measure space, such that $\mu(X)=\infty$. Let $f_n:X \to \mathbb{R}$ be measurable real-valued functions, which converge uniformly to a function $f$. Suppose that $f_n \in L^...
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The set of points with rational coordinates is disconnected

I found the following example: let $Z\subseteq\mathbb{R^2}$ the set of points with rational coordinates. The set $Z$ is disconnected, indeed a separation is given by $\{(x,y)\;|\; x<\pi\}$ and $\{(...
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Give an example of a function that is bounded and continuous on the interval [0, 1) but not uniformly continuous on this interval.

My thoughts was to take $f(x) =\cos(\frac 1x) $ for all $ x \in [0,1)$ as I know this function is continous from $[0,1)$ and is definitely not uniformly continuous as it oscilates non-uniformly. My ...
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On the solvability group

Let $G$ and $H$ be two groups such that $|G|=|H|$; for every natural number $n$ the number of elements of order $n$ in $G$ and $H$ are equal; $H$ is a solvable group. Is $G$ solvable? or Is there ...
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A concrete example of involution in group theory

I am reading the textbook "Introduction to Modern Algebra, Joyce 2017" and in the Cyclic groups and subgroups section, there is a following sentence about involution. An involution $a$ is an ...
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Examples of sequential compact but not compact spaces that do not use ordinals.

I think the title is self explanatory, I'm using Munkres' Second Edition text for Point Set Topology and I can't figure out if such examples are possible.
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1answer
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Hint Predicated on a False Claim?

A while back I asked this question about classifying all noncommutative $p^3$ groups (with $p \ge 3$). The book (see this) I am using gave the following hint ...there is a normal subgroup $N$ of ...
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Counterexample for the following statement

This question arises trying to solve exercise 14H of Willard's General Topology book. That exercise asks us to proof that given any topological space, there exists another space which is Tychonoff ($...
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Spaces in which all closed sets are regular closed

I was reading about the regular closed sets. The definition is Let $X$ be a topological space and $A\subseteq X$. We say that $A$ is a regular closed if $A=\text{cl}(\text{int}(A))$ Then, one ...
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1answer
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If $r(A)=\omega(A)$, is $r(A)=\|A\|$?

Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $F$. It is well know that $$r(A)\leq\omega(A)\leq\|A\|,$$ for every $A\in\mathcal{B}(F)$, where $r(A)$, $\...
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1answer
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Irreducible and aperiodic Markov chain without invariant distribution/measure

Is it possible that a Markov chain is irreducible and aperiodic but without invariant distribution or without an invariant measure? Could someone give examples?
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$\sigma(C_1) \subset \sigma(C_2) \iff C_1 \subset C_2$?

Let $(\Omega, \mathcal{F})$ be a measurable space and $C_1, C_2 \subset \mathcal{P}(\Omega)$. Then, we know that $C_1 \subset C_2 \implies \sigma(C_1) \subset \sigma(C_2)$, where $$ \sigma(A) := \...
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1answer
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Solve: $\|u+v\| \le \|u\| + \|v\|$ with $\|x\| = \left( \sqrt{|x_1|} + \sqrt{|x_2|} \right)^2$

I was given the following task: Check if $x\rightarrow \left(\sqrt{|x_1|} + \sqrt{|x_2|}\right)^2$ is a norm on $\mathbb{R}^2$. I've already shown that $$\|x\| \ge 0\qquad \|x\| = 0 \...
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1answer
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Requirements of remainder in multidimensional Taylor Theorem

Theorem: Let $V$ and $W$ be finite dimensional Banach spaces, $G \subset V$ an open subset, $f: G \to W$ a $n$-times differentiable function and $p \in G$. Then, we have $$ f(x) = \bigg(\sum_{k = ...
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1answer
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Counterexample of $\lim_{x\rightarrow \infty}f(x,t(x))\neq 1$ where $\lim_{x\rightarrow \infty }t(x)=2$.

Given $\lim_{x\rightarrow \infty} f(x,t)=1$ for any fixed t in $(1,3)$, in general $\lim_{x\rightarrow \infty} f(x,t(x))\neq 1$ where $\lim_{x\rightarrow\infty}t(x)=2$ and $t(x)\in (1,3)$ unless we ...
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1answer
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Does closed convex sets having unique nearest points imply the parallelogram law?

It's a well-known result that if $X$ is a Hilbert space, then for any closed convex subset $C$ of $X$, there exists a unique element of $C$ with minimal norm. I'm wondering whether the converse is ...
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Lattice Theory and Topkis Theorem

Topkis Theorem states that: if $f$ is supermodular in $(x,\theta)$, and $D$ is a lattice, then $x^{∗} ( θ ) = \arg\max _{x ∈ D} f ( x , θ )$ is nondecreasing in $θ$. It is not a simple concept for ...
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Homeomorphic spaces are uniformly isomorphic

A continuous function $f$ is a homeomorphism if it is bijective, and open. A uniformly continuous function $f$ is a uniform isomorphism if it is bijective and $f^{-1}$ is uniformly continuous. Is ...
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Are there examples of sequences $(x_n)$ such that $(x_n) \in l^p(\mathbb{N})$ but $(x_n) \not\in l^p(\mathbb{Z})$?

I'm curious about whether/how a bi-directional sequence can have stricter conditions for convergence than a sequence over $\mathbb{N}$. I assume that this is possible, but haven't been able to ...
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1answer
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Uniform inequality for a continuous function

Let $f(x,y)\in \mathcal{C}([a,b]\times[c,d])$ such that $$\exists \xi\in (a,b) : f(\xi,y)\neq 0, \forall y\in [c,d].$$ By the continuity of $f$, we have $$|f(\xi,\cdot)|\geq \min\limits_{[c,d]} |f(\...
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1answer
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Example: limit point compact + $\neg$countably compact+ $\neg$Lindelöf

Question: Find a topological space that is limit point compact (or weakly countably compact) + $\neg$countably compact + $\neg$Lindelöf Notice such a space cannot be $T_1$ (every one-point set ...
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Can you have an infinite descending chain of dual spaces?

If a Banach space $X$ is not reflexive, then you have an infinite ascending chain of (continuous) dual spaces: $X’$, $X’’$, $X’’’$, etc. None of these are isomorphic to each other or to $X$. My ...
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Lattice definition and example

Guys I am struggling to understand the lattice concept: Could you help me with this silly example? Take the collection $\{\emptyset, \{0\}, \{1\}\}$ ordered by inclusion. This is a poset, but not a ...
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1answer
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Example of a matrix where equality occurs for the relation between infinite and 2 norm

Given $A \in \mathbb R^{m\times n}$, I know that: $$\|A\|_2 \le \sqrt {m} \|A\|_\infty$$ $$\|A\|_\infty \le \sqrt {n} \|A\|_2$$ I am supposed to provide an example of a matrix such that the ...
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1answer
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Prove an inequation with $x + y + z + xyz = 4$

$x$, $y$ and $z$ are three positives such that $x + y + z + xyz = 4$. Prove that $\dfrac{x}{y + z} + \dfrac{y}{z + x} + \dfrac{z}{x + y} - x - y - z \le -\dfrac{1}{2}$. When I saw this problem in a ...