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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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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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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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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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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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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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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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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 ...
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 ...
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 ...