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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7
votes
1answer
325 views

Counterexample: a pair of linearly ordered sets that are isomorphic to subsets of the other, but not isomorphic between them [duplicate]

I have encountered myself with the following exercise: Let $\langle A, <_R\rangle$ and $\langle B, <_S\rangle$ be two linearly ordered sets so that each one is isomorphic to a subset of the ...
5
votes
1answer
446 views

Primality test for numbers of the form $N=4 \cdot 3^n-1$

Can you provide proof or counterexample for the claim given below? Inspired by Lucas-Lehmer primality test I have formulated the following claim: Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\...
11
votes
3answers
495 views

Examples of subgroups where it's nontrivial to show closure under multiplication?

Usually when a subgroup is declared, it is trivial (or at least straightforward to a sophomore) to prove that it is a subgroup under multiplication. For example: Homomorphic image and preimage of a ...
13
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11answers
2k views

Examples of non trivial equivalence relations , I mean equivalence relations without the expression “ same … as” in their definition?

Relations defined by formulas such as " x has the same age as y" , " x comes from the same country as y " " a has the same image under function f as b " are obviously equivalence relations, due to the ...
3
votes
2answers
91 views

Partitioning the Reals into two Locally Uncountable, Dense Sets [duplicate]

Is it possible to find two disjoint subsets $X$ and $Y$ of $\mathbb{R}$ such that both are dense in $\mathbb{R}$ and both are locally uncountable? By a locally uncountable set $X \subset \mathbb{R}$, ...
3
votes
2answers
99 views

Is there always a continuous surjection from $H \times G/H$ to $G$?

Question: Is there always a continuous surjection $f: H \times G/H \rightarrow G$? where $G$ is a topological group, $H$ is a subgroup of $G$ and $G/H$ is given the quotient topology. We know $H ...
6
votes
2answers
403 views

How to be good at coming up with counter example in Topology

This is a more generalized question, but does anyone have a set of tips or tricks to come up with distinctive examples and counterexamples in Topology and Analysis? More specific, how can people often ...
2
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4answers
203 views

Examples of contravariant functors

I understand the definition and usefulness of the notion of functor. But I am worrying about the usefulness of the notion of a contravariant functor. Wikipedia writes: There are many constructions ...
3
votes
1answer
46 views

A set $E$ such that $E$ is dense in $[0,1]\times [0,1]$, and the intersection of $E$ and any line parallel to the axes has at most one point

How can I construct a set $E$ such that $E$ is dense in $[0,1]\times [0,1]$, and the intersection of $E$ and any line parallel to the axes has at most one point? I am finding this set in order to ...
1
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0answers
63 views

Example for Lévy's continuity theorem

I am searching a sequence of RV $(X_n)$ for which we prove a convergence in distribution to a random variable $X$, using the fact that the characteristic functions $(\varphi_n)_n$ converges pointwise ...
8
votes
1answer
426 views

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+(...
0
votes
1answer
30 views

Useful bijections [closed]

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) \...
5
votes
1answer
96 views

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 ...
0
votes
2answers
64 views

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 ...
0
votes
1answer
44 views

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 ...
18
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3answers
474 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 ...
2
votes
1answer
105 views

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 ...
0
votes
1answer
41 views

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 ...
1
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2answers
75 views

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 $\lim_{n\to\infty}f_n=f$, $\int f =\lim_{n\to\infty}\int f_n$ ...
3
votes
1answer
111 views

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 ...
1
vote
1answer
55 views

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, ...
8
votes
1answer
140 views

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 ...
0
votes
1answer
71 views

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 ...
3
votes
1answer
59 views

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 ...
0
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2answers
34 views

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 $\lambda \leq 1$ $$...
2
votes
1answer
65 views

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 ...
7
votes
2answers
345 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 ...
2
votes
2answers
58 views

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 ...
0
votes
1answer
31 views

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 ...
0
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1answer
48 views

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 ...
1
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1answer
71 views

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 ...
1
vote
1answer
50 views

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. ...
2
votes
1answer
33 views

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^...
0
votes
0answers
27 views

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 $\{(...
4
votes
2answers
745 views

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 ...
0
votes
0answers
106 views

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 ...
1
vote
4answers
86 views

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 ...
0
votes
1answer
49 views

Give an example of function that satisfies this theorem? [closed]

Give an example of function that satisfies this theorem ? Theorem :The set of points of discontinuity of a monotonic function $f :\mathbb{R} \rightarrow \mathbb{R}$ is at most countable ...
7
votes
1answer
63 views

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.
1
vote
1answer
49 views

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 ...
3
votes
1answer
128 views

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 ($...
4
votes
2answers
510 views

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 ...
2
votes
1answer
26 views

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 $r(A)$, $\...
0
votes
1answer
56 views

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?
1
vote
2answers
43 views

$\sigma(C_1) \subset \sigma(C_2) \iff C_1 \subset C_2$?

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) := \...
1
vote
1answer
47 views

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 \...
4
votes
1answer
79 views

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 = ...
0
votes
1answer
25 views

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 ...
1
vote
1answer
87 views

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 ...
1
vote
0answers
35 views

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 ...