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)

50
votes
30answers
5k views

What are some 'conceptualizations' that work in mathematics but are not strictly true?

I am having an argument with someone who thinks that it's never justified to teach something that is not strictly correct. I disagree: often, the pedagogically most efficient way to make progress is ...
8
votes
8answers
1k views

Polygons with equal area and perimeter but different number of sides?

Let's say we have two polygons with different numbers of sides. They can be any sort of shape, but they have to have the same area, and perimeter. There could be such possibilities, but can someone ...
2
votes
3answers
33 views

Example of ring $R$ with ideals $I\neq J$ such that $R/I \cong R/J$ as modules

It's easy to prove that if $I$, $J$ are two-sided ideals and $R/I\cong R/J$ as modules over $R$, then $I=J$. What about left ideals? Is there a simple counterexample? I believe I've found an answer, ...
0
votes
0answers
20 views

Homomorphic image of intersection equals intersection of homomorphic images?

This is a tangent of this question. I wanted to remark in my answer that it is not generally true that given a group homomorphism $f:G\rightarrow H$ and two subgroups $X,Y\leq G$ that $$f(X\cap ...
4
votes
0answers
43 views

Constructing a family of distinct curves with identical area and perimeter

Two recent questions were posed by Arjuba [1] [2] asking for counter-examples regarding whether two different figures could have the same perimeter and area. Responders quickly raised a number of such ...
2
votes
3answers
555 views

Looking for examples of first countable, compact spaces which is not separable

Could someone give me some classical examples of first countable, compact spaces which is not separable? However, other examples are also welcome. Any help will be appreciated.
0
votes
1answer
31 views

Linear bijection non-preserving Hausdorff propery

My question is: If $f: X \to Y$ is a continuous and linear bijection between topological vector spaces, is it possible that $X$ is Hausdorff and $Y$ is non-Hausdorff? (TVSs are considered in the more ...
1
vote
3answers
42 views

“Full”-Simplification of arbitrary mathematical expressions

I've come across many (classroom) problems, like Roy did, whereby the solution to a problem, $$−3(7−2x)^2−5(1+x)^2$$ is the result of simplifying that expression as much as is possible, i.e. ...
1
vote
0answers
26 views

Separability of conjugacy classes in conjugacy separable semidirect products.

We say that group $G$ is conjugacy separable if for every $g \in G$ the set $g^G = \{cgc^{-1} \mid c \in G\}$ is closed in the profinite topology on $G$, i.e. for every $f \in G \setminus g^G$ there ...
2
votes
1answer
37 views

Semidirect products as (amalgamated) free product

It is well known that $\mathbb{Z}\rtimes \mathbb{Z}_2$ is free product $\mathbb{Z}_2 \star \mathbb{Z}_2$. Are there more examples of these kind?
0
votes
1answer
36 views

Pair of functions $F(x)$ (transcendental),$A(x)$ (algebraic) with expanded series of positive integer coefficient linked by derivative

$$F(x)=\sum_0^{\infty}b_k x^k,b_k\in \mathcal{N} \bigcup 0,\exists M \space b_k \leq M^k$$. $$A(x)=\sum_0^{\infty}a_k x^k,a_k\in \mathcal{N} \bigcup 0,\exists M \space a_k \leq L^k$$ where $F(x)$ is ...
1
vote
1answer
614 views

Example of Converge in measure, but not converge point-wise a.e.?

Can anyone give an exam of Converge in measure, but not converge point-wise a.e.? And also for the converse part, professor asks us to prove "pointwise a.e. implies converge in measure", but think ...
3
votes
2answers
108 views

Is an arbitrary group generated by a traversal of the conjugacy classes?

Let $G$ be a group, and let $\mathcal C$ be the collection of conjugacy classes of $G$. Let $S$ be a traversal of $\mathcal C$ (that is $S$ contains exactly one element from each set in $\cal C$). ...
9
votes
2answers
772 views

compactness / sequentially compact

I'm looking for two examples: A space which is compact but not sequentially compact A space which is sequentially compact but not compact Explanations why the spaces are compact / not compact and ...
5
votes
7answers
184 views

If $A^2 = B^2$, then $A=B$ or $A=-B$

Let $A_{n\times n},B_{n\times n}$ be square matrices with $n \geq 2$. If $A^2 = B^2$, then $A=B$ or $A=-B$. This is wrong but I don't see why. Do you have any counterexample?
3
votes
1answer
38 views

Free cyclic subgroups in a non-abelian group

Is there any non-abelian group $G$ such that for each $a\in G$ and any automorphism $g:\left<a\right>\to \left<a\right>$ the function $$f:G\to G$$ $$f(x) = \begin{cases} g(x) & \text{ ...
3
votes
2answers
271 views

Interesting Problems for NonMath Majors

Sometime in the upcoming future, I will be doing a presentation as a college alumni to a bunch of undergrads from an organization I was in college. I did a dual major in mathematics and computer ...
3
votes
3answers
93 views

Non-separable compact space

Off the top of my head, I can't think of a non-separable compact space. Can you provide a good example?
1
vote
1answer
26 views

If the quotient by the $i$th center is cyclic, does it follow that the original group is abelian?

Let $G$ be a group such that there exists an $i$ such that $G/Z^i(G)$ is cyclic. Does it follow that $G$ is abelian? This question is a generalization of the well known fact that if $G/Z(G)$ is ...
1
vote
1answer
56 views

Let $\int_Xf\,d\mu,\int_Xg\,d\mu$ be finite, must $\int_X fg\, d\mu$ be finite

Let $(X,\scr{M},\mu)$ be a measure space. Let $f,g:(X,\scr{M},\mu)\rightarrow[0,\infty]$ be measurable with finite integrals over $X$. Must $\int_X fg \,d\mu$ be finite ? My guess is yes, I already ...
0
votes
3answers
59 views

Proof by contradiction using counterexample

Why can't we use one counter example as the contradiction to the contradicting statement? Example: Let a statement be A where a-->b. We can prove A is not true by finding a counter example. Now, in ...
11
votes
1answer
110 views

If $f\tau$ is continuous for every path $\tau$ in $X$, is $f:X\rightarrow Y$ continuous?

Let $X$ be a path connected space and $Y$ be a topological space. Let $f:X\rightarrow Y$ be a function such that for every path $\tau:\mathbb{I}\rightarrow X$ , $f\tau:\mathbb{I}\rightarrow Y$ is ...
6
votes
1answer
34 views

Divergence set at radius of convergence

I came up with this question on my own while I was musing around reviewing notes. After unsuccessful Google search (thwarted by a deluge amount of webpages on basic calculus), I decided to ask here. ...
2
votes
0answers
32 views

Natural examples of multiplicatively-graded rings

A ring $A$ is multiplicatively graded if, as an additive group, it decomposes as $$A=\bigoplus_{n\geq 1} A_n$$ and the product satisfies $A_n \times A_m \rightarrow A_{nm}$. A natural example of ...
5
votes
2answers
98 views

Give an example of a function $f$ satisfying $\lim_{x\to 0}(f(x)f(2x))=0$,but $\lim_{x\to 0}f(x)$ does not exists

Question: Give an example of a function $f$ satisfying the condition $$\lim_{x\to 0}(f(x)f(2x))=0$$ and such that $$\lim_{x\to 0}f(x)$$ does not exists. I think this question have many example. But ...
12
votes
2answers
171 views

Do differentiable functions preserve measure zero sets? Measurable sets?

Consider Lebesgue measure on $\mathbb{R}$ and let $f:\mathbb{R}\to\mathbb{R}$ be differentiable. Does $f$ necessarily preserve measure zero sets? Does $f$ necessarily preserve measurable sets? ...
3
votes
1answer
24 views

What are some examples of non-commutative monoids that are both idempotent and self-distributive (on both sides)?

In the presence of the axioms for a commutative monoid, idempotency is equivalent to self-distributivity. Proof. Suppose a commutative monoid is idempotent. Then: $$x(yz) = xxyz = (xy)(xz)$$ On the ...
8
votes
4answers
2k views

Connected metric spaces with at least 2 points are uncountable.

That's a problem I proved (quite a while back) in tiny Rudin. However, I don't really get it. The other questions were actually useful results - I don't think I've ever come near using this result. ...
0
votes
0answers
87 views

Please give me an example of the algorithm where $\Theta$ will be equal to $e^n$

Please give me an example of the algorithm where $\Theta$ or $O$ will be equal exactly to $e^n$ . The algorithm should not be simple counting from 0 till $e^n$ . It should be a clear relation of two ...
253
votes
27answers
15k 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 ...
9
votes
1answer
123 views

Are elements commutative when subgroups are?

Suppose G is a group and for every $H_1,H_2\le G$, $H_1H_2=H_2H_1$. Is $G$ abelian?
0
votes
1answer
40 views

Example of a function $f$ which is Lebesgue integrable on $[0,1]$, but max{$f,0$} is NOT Lebesgue integrable?

Can anyone come up with an example of a function $f$ which is Lebesgue integrable on $[0,1]$, but max{$f,0$} is NOT Lebesgue integrable? Thanks.
0
votes
0answers
19 views

A Lindelof non-scattered space $X$ which is not an extention of $\mathbb R$

Is anyone familier with an example for a Lindelof non-scattered topological space $X$ which is not an extention of $\mathbb R$ (with Euclidean topology). I am looking for an example which is not a ...
4
votes
3answers
71 views

Non-isomorphic algebraic structures such that each surjects homomorphically onto the other

Off the top of my head, I cannot think of any algebraic structures $X$ and $Y$ such that each surjects homomorphically onto the other, yet $X$ and $Y$ are non-isomorphic. What are some examples of ...
0
votes
1answer
66 views

An example in Spivak's Calculus on Manifolds (chain rule).

Spivak gives an example which has step that is giving me some problems to get it, even if it's supposed to be trivial. Spivak says: Let $f:\mathbb{R}^2\to\mathbb{R}$ given by $f(x,y)=\sin(xy^2)$. ...
2
votes
2answers
49 views

Convergence of integrals but $\int_a^b|f_n(x)-f(x)|dx$ does not converge to $0$

Can I have an example of a sequence of continuous functions $(f_n)_n$ and a continuous function $f:[a,b]\to \mathbb{R}$ such that $$\int_a^bf_n(x)dx\to\int_a^bf(x)dx,$$ when $n\to+\infty$, but ...
2
votes
1answer
68 views

An operator such that $\|A\|^2 \neq \|A^2\|$

The question asks for a bounded linear operator on a Hilbert space satisfying the condition in the title. This is what I came up with: Let $A_1:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be a 90-degree ...
-1
votes
1answer
52 views

Why set is not equal its closure minus its boundary? [closed]

Why $ \Omega \neq \bar{\Omega} \setminus \partial \Omega $ ? Can somebody show any counterexample?
6
votes
1answer
92 views

Non-examples for the Kato-Rellich Theorem

The Kato-Rellich Theorem is a classical result stating that if $A,B$ are unbounded operators on a Hilbert space with $A$ self-adjoint, $B$ symmetric, $\mathcal D (A)\subset \mathcal D(B)$ and $$ ...
3
votes
2answers
52 views

Groups reluctant to have infinite subgroup

Is there a group with only one infinite subgroup‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌?
0
votes
1answer
35 views

A counter-example to Dini's Theorem (after removing a hypothesis)

Recall Dini's Theorem: Let $K$ be a compact metric space. Let $f: K\to\mathbb{R}$ be a continuous function and $f_{n}: K\to\mathbb{R}$, $n\in\mathbb{N}$ be a sequence of continuous functions. ...
6
votes
1answer
48 views

Definition of the normalizer of a subgroup

Let $G$ be a group and $H$ a subgroup of $G$. Is there any counterexample to the assertion $N_G(H):=\{g\in G\mid gHg^{-1}=H\}=\{g\in G\mid gHg^{-1}\subset H\}$? Thanks!
1
vote
2answers
90 views

Conjecture: if $a+b$ and $ab$ are rational, $a$ and $b$ are rational

I can't find a rigorous proof but I have a feeling it's true. Informal argument: Suppose $a+b$ and $ab$ are rational, $a$ and $b$ are irrational (since just one can't be irrational). Then $a$ and ...
2
votes
1answer
46 views

Fundamental theorem of calculus necessary assumption

The fundamental theorem of calculus is stated as follows: Let $f$ and $F$ be real-valued functions defined on a closed interval $[a, b]$ such that the derivative of $F$ is $f$. That is, $f$ and $F$ ...
1
vote
2answers
54 views

Sequentially compact space

Is every sequentially compact space metrisable? If not, then, can you give me an example of a sequentially compact space that is not compact.
2
votes
1answer
48 views

Can any “relevant” topological spaces be decomposed into an uncountable product?

Can any "relevant", as meaning generally useful topological spaces be decomposed into an uncountable product of other topological spaces with the product topology? Many thanks in advance.
3
votes
1answer
33 views

Example of a function f that is Generalized Riemann Integrable, but its square is NOT Generalized Riemann Integrable.

I am reading a section about Generalized Riemann Integral (Kurzweil-Henstock), and there was a problem on that section to provide an example of a function $f$ on $[0,1]$ that is Generalized Riemann ...
1
vote
1answer
73 views

An example of when a product in a category may not exist.

My question is about the category of finitely generated Abelian groups; in particular, I want to show, by definition, that there exists a set of objects $T$ in this category for which there is no ...
1
vote
1answer
64 views

Closed 4-manifolds with uncountably many differentiable structures

I know that $\mathbb{R}^4$ admits uncountably many differentiable structures and I was wandering what happen if we consider closed (or just compact) 4-manifolds. Are there any closed (or compact) ...
3
votes
0answers
75 views

A sequence of subsets of an infinite group.

Is there an infinite group $G$ such there is not any sequence $(A_n)$ of its subsets such that always $$A_n=A_n^{-1}, \quad A_{n+1}A_{n+1}\subsetneqq A_n$$ ?