Mathematics Stack Exchange
Mathematics Stack Exchange

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.

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 in conjunction with another tag to clearly specify the subject.

5506 questions
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What's a very simple example of an equivariant map?

If we take $S^1$ acting on $S^2$ by rotation, then the height function $h: S^2\to R$ is an example of an invariant map (or an equivariant map where the action on $R$ is the trivial one). I'm looking for an easy example of an equivariant map which is…
slackenerny
  • 625
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1 answer

Examples of "finite character"

The property of being a linear ordering has finite character, i.e. a relation linearly orders a set if it linearly orders all of its finite subsets. This is a trivial corollary of the definition. The property of being linearly independent has…
Michael Hardy
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5 answers

Examples of solving a problem by introducing new parameters

First let's see some examples: I've seen a lot of examples of calculating the definite integrals by introducing a suitable parameter, but still cannot understand why this method so effective, while calculating the integral directly could be…
painday
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1 answer

Examples of rational equivalence.

I have to do a lesson (1 h.) about a very basic introduction to intersection theory. In order to do this I'd like to find a way in order to examplain the concept of rational equivalence giving a lot of examples. Could you suggest me examples of…
ArthurStuart
  • 4,932
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5 answers

Looking for a theorem that can be reduced to a particular case

I am looking for a theorem that can be reduced to a particular case, i.e., such that there is a particular case for which if the theorem holds, then this easily implies its general validity. About the context: I am writing an article about the Four…
Alma Arjuna
  • 3,759
3
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1 answer

Example of a strictly convex function unbounded in $\mathbb{R}$

Is there some strictly convex function defined in $\mathbb{R}$ to be unbounded(above and lower)? For example, $f:(\infty,0]\to \mathbb{R},$ $f(x)= -x^2$ is a strictly convex function. However, this function is not defined in all $\mathbb{R}$ and…
MAOC
  • 616
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Function which takes on every value in $[0,1]$ an infinite number of times.

Give an example of a continuous real-valued function $f$ form $[0,1]$ to $[0,1]$ which takes on every value in $[0,1]$ an infinite number of times. My example goes like this: Take $f(x)=\lim_{n \to \infty} g(x)$ where.…
zaemon_23
  • 465
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Bijective and smooth function $\mathbb{R} \to \mathbb{R}$

I am a tutor for complex analysis this semester and my students had to show as an exercise that every holomorphic and bijective function $f: \mathbb{C} \to \mathbb{C}$ is affine linear, i.e. there exist some $a, b \in \mathbb{C}, a \neq 0$ such that…
kade
  • 121
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0 answers

product σ-algebra counter example

In Real Analysis by Folland, the product σ-algebra $ \displaystyle \bigotimes_{\alpha \in A} \mathcal{M_{\alpha}} $ on $X$ is defined by the σ-algebra generated by $ \{ \pi ^{-1} _{\alpha} (E_{\alpha}): E_{\alpha} \in \mathcal{M_{\alpha}} \} …
Hana Kim
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Examples of functions f(x,y)

i need to find examples of functions $f(x,y):x,y $ belong to $R^2$ such that: $f$ is constant on the parabolas $y=c(x^2+1)$ c belongs to $R$. And the second one should be bounded on the unit circle, but not bounded on $x^2+y^2<4$. Any suggestions?
CodeHoarder
  • 519
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1 answer

Prove using the smallest counterexample technique that: $\binom {2n}n\leqslant4^n.$

Actually what i know is that i must assume that this statement is false and then try to come up with non sense statement. Prove by the smallest counterexample technique the statement $$\binom {2n}n\leqslant4^n.$$
Amirpasha
  • 49
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smooth in one direction, nowhere differentiable in the other

Can anyone think of an example $u\in L^2(\mathbb{R}^2)$ such that $u$ is smooth in $x_1$ but nowhere differentiable in $x_2$?
Karo
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Finding a counterexample

Why is this wrong? Doesn't $a | b$ mean $a$ divides $b$ so $b = a(x)$? So then if $a=3, b=4, d=7$ then $d | ab$ will mean $12 = 7(12/7)$, $d | a$ will mean $3 = 7(3/7)$, $d | b$ will mean $4 = 7(4/7)$ so $d | ab$ is not equal to $d | a$ or $d |…
user30200
  • 67
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Any Literature Related To Search of Counter Examples?

I am looking for the well-summarized literature which holds any information about how to understand the overall logic related to searching of Counter examples. I think,heuristically,this knowledge quite important and much more foundational because…
Beverlie
  • 2,645
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amenability on Banach algebras

we know that $C_0(G)$ is Banach $M(G)$ bi-module for locally compact group $G$. I would like to know that is there derivation of $M(G)$ into $C_0(G)$? Or is it inner every derivation of $M(G)$ into $C_0(G)$ ?
Mehrdad shabani
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