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 ...

learn more… | top users | synonyms (2)

12
votes
0answers
35 views

Does $A^2 \cong B^2$ imply $A \cong B$ for rings?

If $A$ and $B$ are two unital rings such that $A \times A \cong B \times B$, as rings, does it follows that $A$ and $B$ are isomorphic (as rings)? I believe that the answer is no, but I can't come ...
0
votes
2answers
38 views

constructing a specific (real-) analytic function

Im searching for an example of a special-behaving analytic function. Maybe you can beat me to constructing such one. The criterias are $g :\mathbb{R}\rightarrow \mathbb{R^+}$ is analytic $g$ is $\...
1
vote
1answer
50 views

In which topologies do open sets maintain open under countable or arbitrary intersection?

We know that in the usual topology, countable or arbitrary intersection of open sets can zoom into a singleton, hence is not in the topology. I am curious if there is well known classes of ...
-2
votes
1answer
46 views

Not Abelian group G with Z(G) that contains only two elements? [on hold]

Is there an example of which is not Abelian group G, and Z(G) contains only two elements?
1
vote
3answers
49 views

Intermediate value property with no continuity

Definition: A real function f has the intermediate value property on an interval I containing [a,b] if f(a) < v < f(b) or f(b) < v < f(a); that is, if v is between f(a) and f(b), there is ...
-3
votes
1answer
40 views

Formal definition of “proexample”. [on hold]

Where in the literature do we find the preferred formal definition of “proexample” as in: the number zero is a proexample for the existential sentence "some integer is neither positive nor negative"? ...
5
votes
1answer
47 views

What is a measurable set $E \subset [0, 1]$ such that the closure of $E$ is $[0, 1]$ and $m(E) = \epsilon$?

Suppose $\epsilon \in (0, 1)$ and $m$ is Lebesgue measure. What is a measurable set $E \subset [0, 1]$ such that the closure of $E$ is $[0, 1]$ and $m(E) = \epsilon$?
1
vote
1answer
36 views

If 2 loops with equal base points are homotopic, must they be homotopic relative to the base point?

Let $X $ be a topological space and $\mathbb {S}^1$ be the set of complex numbers with magnitude 1 equipped with the inherited topology from $\mathbb {C} $. If we have 2 loops $f,g:\mathbb {S}^1\...
4
votes
0answers
63 views

If $f$ is defined on $\mathbb{R}$ and $f$ is unbounded, is it necessarily true that $\lim_{|x|\to\infty} |f(x)| = \infty$?

This question comes from the following problem: A real-valued function $f$ defined on $\mathbb{R}$ has the following property: For every positive $\epsilon$, there exists positive $\delta$ such ...
-4
votes
0answers
30 views

Solve for the conditions given below [closed]

\begin{align} f(c, d)&= a;\\ g(c, d)&= b;\\ h(a, b, c)&= d. \end{align} The functions $f$, $g$, $h$ are defined for all $a,b,c,d\in\mathbb R$. For instance: $h$ can be Division; $a$, $b$, ...
1
vote
2answers
38 views

Useful analogy to interpret the notion of evolutionary stable strategy (ESS)

I am seeking a good analogy to understand the concept of evolutionary stable strategy (state) Let $\pi$ denote the fitness of a population, $\pi_{ij}$ is the fitness of strategy $i$ against strategy $...
6
votes
5answers
905 views

Formal definition of “counterexample”.

What is the preferred formal definition of “counterexample” as in: zero is a counterexample for "every integer is either positive or negative". Where in the literature is the notion of “counterexample”...
2
votes
0answers
30 views

What is an example of lower semicontinuous functions not satisfying this?

Let $X$ be a locally compact Hausdorff space and $\mu$ be a Radon measure on $X$. Let $u:X\rightarrow [0,\infty]$ be a lower semicontinuous function such that $\int_X u d\mu <\infty$. Then, does ...
0
votes
2answers
44 views

Examples of $R$-modules $X$ such that $(X \setminus TX) \cup \{0\}$ isn't a submodule.

Work over an ambient commutative ring with unity. Given a module $X$, write $TX$ for its submodule of torsion elements. Suppose we want to find the "submodule" of torsion-free elements of $X$. So ...
0
votes
0answers
31 views

Example of function

Consider $O(2)$, the orthogonal group in dimension 2, $C_{k}$ the cyclic group of order $k$ and $D_{k}$ the dihedral group of order $2k$. Is it possible to have an example of function $f\in L^{\infty}(...
20
votes
14answers
2k views

How are proofs formatted when the answer is a counterexample?

Suppose it is asked: Prove or find a counterexample: the sum of two integers is odd The fact that 1 + 1 = 2 is a counterexample that disproves that statement. What is the proper format in which ...
2
votes
1answer
28 views

If $f:X\to Y$ is continuous and $X$ is a totally bounded metric space, is $f(X)$ also bounded?

If $X$ and $Y$ are metric spaces and $X$ is totally bounded and $f:X\to Y$ is continuous (not necessarily uniform), is it true that $f(X)$ is also bounded? How can we prove it or is there any counter ...
0
votes
0answers
77 views

Counterexample 3n + 1 problem (Collatz) Exponential and linear Diophantine equation

So, I have found a sufficient condition (not necessary) for finding a counterexample to the 3n + 1 problem, namely the existence of solutions for the following two-parameter family of Diophantine ...
2
votes
1answer
37 views

Counterexamples about locally compact sets on the real line

Is there a counterexample in the space $\mathbb{R}$ with it's usual metric to the statements: The union of two locally compact subsets of $\mathbb{R}$ is locally compact The complement of a locally ...
3
votes
0answers
35 views

On the right adjoints of inverse image functors ($f^* \dashv \forall_f$)

Given is an ambient category $\mathcal{A}$ with finite limits. For the remainder of this post, a subobject of an object $A$ is a mono $m : M\to A$ and $\operatorname{Sub} A$ is the preordered set (/...
2
votes
1answer
24 views

Discontinuous (no continuous representative) function $u \in W^{1,p}(\Omega)$?

For an open set $\Omega \subset \mathbb{R}^N$ and $p \leq N$ we know that there are functions in $W^{1,p}(\Omega)$ which don't belong to $L^{\infty}(\Omega)$. We also know that for $p>N$ there is a ...
2
votes
2answers
64 views

Suppose $ab\equiv 0 \pmod{n}$, and that $a$ and $b$ are positive integers both less than $n$. Does it follow that either $a | n$ or $b | n$?

Suppose $ab\equiv 0 \pmod{n}$, and that $a$ and $b$ are positive integers both less than $n$. Does it follow that either $a | n$ or $b | n$? If it does follow, give a proof. If it doesn’t, then give ...
4
votes
0answers
41 views

Obvious application of internal category theory outside from topoi

The nLab lists a bunch of examples for internal categories in various categories. If we think of a topos as a "universe" for mathematics the need for internal categories in a topos becomes obvious. ...
0
votes
2answers
51 views

An example of subset $A$ such that $A \cap K$ is open in $K$ for each compact set $K$, but $A$ is not open. [duplicate]

Let $X$ be a topological space. For any $A \subseteq X$, consider two possible conditions on $A$: 1) $A$ is open in $X$; 2) $A \cap K$ is open in $K$, for each compact set $K \subseteq X$. Then $(...
1
vote
1answer
73 views

Counterexamples in Analysis

I want to (dis)prove the following statement: A sequence of functions which converges almost uniformly implies uniform convergence for that sequence of functions. I'm sure I've read up on a ...
6
votes
2answers
80 views

Noteworthy examples of finite categories

So far all the finite categories I have encountered fall into one of these c̶a̶t̶e̶g̶o̶r̶i̶e̶s̶ sets: finite monoids finite preorders just formal devices to explain, what a "diagram" in another (...
1
vote
0answers
23 views

Example of Non-Measurable Sets in Product Space

If $\mu$ and $\nu$ are measures on $X$ and $Y$, is there an example of a set $E\subset X\times Y$ such that $E_x,E^y$ are measurable for all $(x,y)$ but $E$ is not measurable with respect to $\mu\...
1
vote
1answer
21 views

Simple example of a function which is in $W^{1,p}(\Omega)$ but not in $L^{\infty}(\Omega)$?

I am looking for a simple (intuitive) example of a function $u \in W^{1,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$ is an open set and, obviously, $p \leq N$. Sobolev embedding theorem asserts ...
0
votes
1answer
57 views

Regular topological spaces need not to be normal

I was looking for a counterexample for the following statement: "A regular topological space need not to be normal." I don't understand how to use the lemma to prove Theorem 7: http://fac.hsu.edu/...
0
votes
4answers
45 views

Find a counter example

The interior of the union is the union of the interiors. $\text{int}\left(A\cup B\right) = \text{int}(A) \cup \text{int}(B)$ I'm not too sure about to get started with this one. Any hints so as to ...
0
votes
0answers
19 views

Basis Function Algorithm, In The NURBS book

On page 74, Peigl explained an algorithm about computing a single basis function. first lines of this algorithm are handling some special cases. ...
1
vote
0answers
33 views

Reference request: how to check whether a set is invariant for a second order dynamical system?

I am looking for some examples where invariant set is proved for second order systems For a example, consider the Van Der Pol equation: $$\dot x_1 = x_2$$ $$\dot x_2 = -x_1 + 0.5(1-x_1^2)x_2$$ In ...
1
vote
3answers
87 views

Are there any examples of Graphs in nature? [closed]

When it comes to fractals, there are several examples we can point to and say 'this is a fractal', such as snowflakes, ferns, trees and coastlines. Are there any equally clear examples of graph and ...
8
votes
1answer
80 views

Is there a measure space $(X,\mathcal M, m)$ such that $\{m(E) \mid E \in \mathcal M\} = \Bbb Q_{\geq 0} \cup \{+\infty\}$?

I have in mind the following question: Is there a measure space $(X,\mathcal M, m)$ such that the range of $m$ satisfies $S:=\{m(E) \mid E \in \mathcal M\} = \Bbb Q_{\geq 0} \cup \{+\infty\}$? (...
1
vote
1answer
46 views

If $f(x,y)$ is a function that its contour lines are straight, is it necessary looks like $f(x,y) = ax + by + c$

If $f(x,y)$ is a function that its contour lines are straight, is it necessary looks like $f(x,y) = ax + by + c$? Well, in the answer is no. it is written that $e^{x+y}$ for every $(x,y)$ has ...
3
votes
1answer
56 views

Is there a searchable database of mathematical objects that you can search by property?

For example, I could search for functions that are continuous, but that don't have differentiability, and come up with a continuous non-differentiable function. Or a smooth but non-analytical function....
4
votes
2answers
93 views

Is boundedness required in equivalence between $\frac1n\sum_{k=1}^na_k\to0$ and $\frac1n\sum_{k=1}^na_k^2\to0$?

Suppose $a_n$ is a sequence of non-negative real numbers. If $a_n$ are un-bounded, then I want to know if $\dfrac{1}{n}\sum_{k=1}^na_k\to0$ as $n\to\infty$ is equivalent to $\dfrac{1}{n}\sum_{k=1}^...
1
vote
1answer
24 views

Example of nonuniqueness of asymptotes of a ray

Let $(M, g)$ be a complete Riemannian manifold and let $\gamma : [0, \infty) \to \mathbb{R}$ be a ray, i.e. a unit speed geodesic such that for every $s, t \ge 0$ : $$ dist\big(\gamma(s), \gamma(t)\...
0
votes
0answers
35 views

Are there groups $G$ isomorphic to $\mathrm{Aut}\left( \mathrm{Aut} (G)\right)$, such that $G\not\cong \mathrm{Aut}(G)$?

Trivial examples for the first condition are easy to find: $G={1},C_2$. Are there any (finite\non-finite) groups that satisfy the conditions in the title?
2
votes
1answer
54 views

Define any non-commutative operation for the group $\left({\mathbb{R}, \circ}\right)$ [closed]

Let $\mathbb{R}$ denote the set of real numbers. Given that $\left({\mathbb{R}, \circ}\right)$ is a group, provide any definition for $\circ$, so that $\circ$ is not commutative.
10
votes
3answers
1k views

A binary operation, closed over the reals, that is associative, but not commutative

I am aware that matrix multiplication as well as function composition is associative, but not commutative, but are there any other binary operations, specifically that are closed over the reals, that ...
-1
votes
4answers
70 views

Disprove the statement: If $g\circ f=I_X$then $f\circ g=I_Y$. [closed]

If someone could walk me through this I would greatly appreciate it. Disprove the following statement: Let $f : X \rightarrow Y$ and $g : Y \rightarrow X$ be functions. If $g\circ f=I_X$...
4
votes
1answer
109 views

are continuous functions that map measure zero sets to measure zero sets absolutely continuous?

Let $I$ be a closed interval and $f:I\rightarrow \mathbb{R}$ be a continuous function which maps measure zero sets to measure zero sets. If $f$ is monotonically increasing, $f$ must be absolutely ...
2
votes
1answer
29 views

If a sequence converges in measure, are convergent subsequences of it all converge to the same limit?

Let $f_n:X\rightarrow \mathbb{C}$ be a sequence of measurable functions such that $f_n\rightarrow f$ in measure. Let $f_{n_k}$ be a subsequence of $f_n$ such that $f_{n_k}\rightarrow g$ pointwise a.e....
11
votes
4answers
1k views

Non-abelian group with infinitely many abelian subgroups

I'm looking for a non-abelian group which has infinitely many abelian subgroups. Do you know any examples of such groups?
3
votes
4answers
112 views

Disprove the statement $f(A \cap B) = f(A) \cap f(B)$ [duplicate]

If someone could walk me through this I would greatly appreciate it. Disprove the following statement: If $f : X \rightarrow Y$ is a function and $A$, $B$ are subsets of $X$ then $f(A \cap B) = f(A)...
1
vote
2answers
31 views

Why can projection function on $X \times S$ be regarded as a local homeomorphism?

I am studying some properties of local homeomorphism I am in particular trying to find a local homeomorphism that is not a homeomorphism and the projection function seems to be the perfect candidate ...
2
votes
1answer
77 views

Domain Monotonicity - Neumann eigenvalue problem (Edit)

Related to the question : http://mathoverflow.net/questions/242136/why-m-1-subset-m-2-not-rightarrow-n-m-1-lambda-leq-n-m-2-lambda The Neumann eigenvalues of the rectangle with sides $a$ and $b$ are $...
1
vote
1answer
35 views

Does there exists two way embedding between two non-homeomorphic spaces?

I am searching for a two way embedding between two non-homeomorphic spaces. In other words, I want two non-homeomorphic spaces such that $X$ is embedded in $Y$ and $Y$ is embedded in $X$. Recall ...
1
vote
1answer
29 views

Reference request: product Borel $\sigma$-algebra of non-separable metric spaces

The following is a proposition in Folland's Real Analysis about product sigma algebra: Here $\mathcal{B}_X$ denotes the Borel $\sigma$-algebra on $X$. Could anyone come up with an example that ...