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 would be a counterexample if hypothesis of mean value theorem is slightly changed?

Let $f:[a,b] \rightarrow \mathbb{R}$ be a function which is continuous on $(a,b]$ and differentiable on $(a,b)$. Is there any function such that $f(b)-f(a)≠(b-a)f'(x), \forall x\in (a,b)$? There was a typo, and now it's edited. I wanted to know…
Katlus
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does there exist a compact, contractible subset of $\mathbb R^2$ that cannot be dissected into $4$ parts of equal area by perpendicular lines?

There was a question asked here, to show that a convex subset of $\mathbb R^2$ can be cut into $4$ pieces of equal area. In an attempt to prove it, I tried to more or less apply a "ham sandwich approach" twice and deduce the answer. The problem was…
Andres Mejia
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Is there any function in $L^2$ that is not integrable?

I know that there are functions in $L^2$ that are integrable but not continuous. Is there any function in $L^2$ that is not even integrable?
3x89g2
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Ring subset which absorbs but is not an additive subgroup

Are there any undegrad-level examples of ring subsets which possess absorbtion property (as in ideal definition) but are not ideals (i.e. are not additive subgroups)?
Artem Pelenitsyn
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Can someone give example(s) of projective non-singular algebraic varieties over $\mathbb{C}$?

I may know that the term "projective" relates to kind of virtual added points at infinity of the variety. I may know that the term "non-singular" relates to the fact that the variety has no singular points and that it might have a link with an idea…
someone
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Counter-example of an affirmation

Does anyone know any example that invalidates the following affirmation: If a morphism $f:A\to A$ induces the identity $\hat f:\operatorname{Spec} \left( A \right) \to \operatorname{Spec} \left( A \right)$ then $f = \operatorname{id} _A $.
user86188
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Example of a map from the natural numbers to the positive rational numbers less than 1

I want an example of a bijective function $f:\mathbb{N} \to \{n \in \mathbb{Q}:0
Rounak Sarkar
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Examples about strictly concave functions with $f(x+z)=f(x)+f'(x)z$

I just learned that a continuously differentiable function $f$ is strictly concave if $f(x+z)
zz273
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Example of Bijection Function from Z+ X Z+ to N X ODDz

I was wondering if anyone could give me some ideas for a bijective function from $Z^{+} \times Z^{+} \rightarrow N \times O D D$ (Natural number include 0 in this case.) One idea I had was to define as $ f(x)= gcd(a, b),$ if $x=2^{a} b,$ b odd. But…
yi G
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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 $\{(x,y)\;|\;x>\pi\}$. Question. Why? Thanks!
Jack J.
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Functions that take out the inverse operation

Are there are there any real-values functions $f(t)$ other than $f=t,1/t$ and $\pm 1$ such that $f(t)f(t^{-1})=1$ for $t>0$?
Andrew Yuan
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Counterexample on the limit of $\frac{f(x)}{x}$

Is the following statement true or false? $f$ is defined on the set of all real numbers, such that $\lim \limits_{x\to 0} \dfrac{f(x)}{x}$ is a real number $L$ and $f(0)=0$. Then $L=0$? I can't draw up any counterexample. Would be grateful for hint.
RFZ
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Fermat little theorem example

I've just learned about Fermat's little theorem and doing some examples. If prime number P is 31 and the integer number a is 11 then the residual will be 25. Where am I wrong? Thanks
Firooz Salami
  • 11
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Counterexample for simple statement

I am refreshing my math skills by reading through one of my math books: Mathematik, the second edition. In Exercise 2.8, one question was if $\forall (x,z) \in \mathbb{R}^2 \,\, \exists y \in \mathbb{R}:x\cdot y = z $ is a true statement. In the…
Xlaech
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Bounded distance generating function

There is a distance-generating function $\omega(x) : X \rightarrow R $ in the definition of Bregman distance, which is then defined as $V(x,y) = \omega(y) − \omega(x) − \langle\omega^\prime(x), y − x\rangle$. One possible example of $\omega(x)$…
Andrei Kulunchakov
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