What is Counterexample: Definition and 55 Discussions

In logic (especially in its applications to mathematics and philosophy), a counterexample is an exception to a proposed general rule or law, and often appears as an example which disproves a universal statement. For example, the statement "all students are lazy" is a universal statement which makes the claim that a certain property (laziness) holds for all students. Thus, any student who is not lazy (e.g., hard-working) would constitute a counterexample to that statement. A counterexample hence is a specific instance of the falsity of a universal quantification (a "for all" statement).In mathematics, the term "counterexample" is also used (by a slight abuse) to refer to examples which illustrate the necessity of the full hypothesis of a theorem. This is most often done by considering a case where a part of the hypothesis is not satisfied and the conclusion of the theorem does not hold.

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  1. M

    Can anyone please verify/review this counterexample?

    Disproof: Here is a counterexample: Suppose p+a^2=25, where 25 is a positive integer. Then we have 25=0+25 =9+16 =16+9...
  2. vibha_ganji

    Finding a Counterexample to a Wrong Statement about Limits

    I’m complete stuck on this problem. I am not sure how to start to find a counterexample to this statement.
  3. L

    A Same open sets + same bounded sets => same Cauchy sequences?

    Let ##d_1## and ##d_2## be two metrics on the same set ##X##. Suppose that a set is open with respect to ##d_1## if and only if it is open with respect to ##d_2##, and a set is bounded with respect to ##d_1## it and only if it is bounded with respect to ##d_2##. (In technical language, ##d_1##...
  4. J

    MHB Resolution method and counterexample

    Hi! Am new to this forum, but I have looked around here for some time now, since am studying a course of logic in the context of computer science. I have a very important exam in a few days, and while I thought I got it, I got shocked when I was looking on previous graded exams to see what I...
  5. G

    I Counterexample to the Poynting theorem

    The counter-example is as follows: We have a rectangular toroid ferrite(ring ferrite), magnetized in a closed loop around the ring. We put capacitor plates on top and bottom surfaces, with suitable direction. Now the Poynting vector points inwards or outwards. We look at a cylindrical surface...
  6. SSequence

    I Counterexample Required (Standard Notations)

    I came up with the following qualification condition (for which I am asking for a counterexample). Some Background: A notation can be thought of as a mapping from an ordinal p∈ωCK to a subset of natural numbers. Generally there can be a few variations it seems: -- one can decide whether to use...
  7. dkotschessaa

    A A pasting lemma counterexample (of sorts)

    From the pasting lemma we have that if ## X = A \cup B ## and ##f: A \rightarrow Y ## and ## g: A \rightarrow Y## are continuous functions that coincide on ## A \cup B ##, they combine to give a continuous function ## h: X \rightarrow Y ## s.t. ## h(x) = f(x) ## for ## x \in A ## and ## h(x) =...
  8. micromass

    Challenge Aren't you tired of counterexamples already?

    And we continue our parade of counterexamples! Most of them are again in the field of real analysis, but I put some other stuff in there as well. This time the format is a bit different. We present 10 statements that are all of the nature ##P## if and only if ##Q##. As it turns out, only one of...
  9. micromass

    Challenge Yet another counterexample challenge

    Well, the last thread of counterexamples was pretty fun. So why not do it again! Again, I present you a list with 10 mathematical statements. The only rub now is that only ##9## are false, thus one of the statements is true. Provide a counterexample to the false statements and a proof for the...
  10. micromass

    Challenge Micromass' big counterexample challenge

    I adore counterexamples. They're one of the most beautiful things about math: a clevery found ugly counterexample to a plausible claim. Below I have listed 10 statements about basic analysis which are all false. Your job is to find the correct counterexample. Some are easy, some are not so easy...
  11. Math Amateur

    MHB UFDs .... Counterexample - I_8 = Z/8Z

    I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ... I am currently focused on Section 3.6 Unique Factorization ... I need help with an aspect of Example 3.70 ... The relevant text from Rotman's book is as...
  12. K

    MHB Direct sum of p-primary components of an R-module counterexample?

    Let $x \in R - \{0\},$ where $R$ is a domain. Define $T_x(M) = \{m \in M \ | \ x^n m=0 \ \ \mathrm{for \ some} \ n \in \mathbb{N}\}$ as the $x$-torsion of $M.$ I know that $T_x(M \oplus N) = T_x(M) \oplus T_x(N)$ for $R$-modules $M,N$ only if $R$ is a PID. But I can't think of a...
  13. E

    MHB Solving for $\theta$: Which Value is a Counterexample?

    Which value for $\theta$ is a counterexample to sin^2$\theta$+cos^2$\theta$=tan^2$\theta$ as an identity? a) pi/4 b) 5pi/4 c) pi/3 d) It is an identity So I tried subbing in each value (a, b, c) in as x and then finding the exact value from that but I'm not getting it.
  14. C

    Looking for counterexample in inequality proof

    Hi guys, I have to teach inequality proofs and am looking for an opinion on something. Lets say I have to prove that a2+b2≥2ab. (a very simple example, but I just want to demonstrate the logic behind the proof that I am questioning) Now the correct response would be to start with the...
  15. T

    Looking for a Counterexample to show that a PowerSet is False

    Homework Statement I am trying to find a counterexample to show why the below statement is False! ρ = PowerSet since I couldn't find the symbol. ρ(A x B) = ρ(A) x ρ(B) Homework Equations N/A The Attempt at a Solution Aside from googling for three days. I read/reread my...
  16. I

    MHB Find Counterexample for Expression about Limit of Composition Function

    Suppose that $U$ is open in $\mathbb{R}^{m}$, that $L\in U$ and that $h:U\setminus \left \{ L \right \}\rightarrow \mathbb{R}^{p}$ for some $p\in N$. If $L=\lim_{x\rightarrow a}g(x)$ and $M=\lim_{y\rightarrow L}h(y)$. Then $\lim_{x\rightarrow a}(h\circ g)(x)=M$. (Someone told me that this...
  17. A

    Is Correlation Coefficient an Informative Indicator in Real-World Datasets?

    Hi, Are you aware of any dataset (in R or elsewhere) consisting of a sample from two variables where the correlation coefficient is (approximately) equal to 1, but the variables refer to completely irrelevant things, i.e. one measuring something that happens on Earth and the other something...
  18. M

    Counterexample where X is not in the Lebesgue linear space.

    Example where X is not in the Lebesgue linear space. Homework Statement I'm trying to find an example where \lim_{n \to +\infty} P(|X|>n) = 0 but X \notin L where L is the Lebesgue linear space. Relevant equations: X is a random variabel, P is probability. I is indicator function. The...
  19. M

    Counterexample where X is not in the Lebesgue linear space.

    I'm trying to find a counterexample where \lim_{n \to +\infty} P(|X|>n) = 0 but X \notin L where L is the lebesgue linear space. ∫|X|I(|X|>n)dp + ∫|X|I(|X|≤n)dp = ∫|X|dp therefore ∫nI(|X|>n)dp + ∫|X|I(|X|)dp ≤ ∫|X|dp Suppose ∫I(|X|>n)dp = 1/(n ln n) Clearly the hypothesis is satisfied...
  20. A

    Normal matrix that isn't diagonalizable; counterexample?

    I've been reading that the diagonalizable matrices are normal, that is, they commute with their adjoint: ##M^*M=MM^*##, where ##M^*## is the conjugate transpose of ##M##. So a matrix is diagonalizable if and only if it is normal, see: http://en.wikipedia.org/wiki/Normal_matrix But from...
  21. P

    Help Me Prove this Identity (or find a counterexample)

    Let f be an analytic function defined in an open set containing the closed unit disk and let z in ℂ be fixed. I've simplified a more complicated expression down to this identity, and as implausible as it looks, after some numerical checking it does in fact appear to be true, but I can't find a...
  22. 1

    Help finding a counterexample for a relation's transitivity

    Homework Statement I'll spare most of the details, R = {(x,y) | |x-y| < 5} I need to find a counter-example to show that it is not transitive. I'm having trouble. The Attempt at a Solution First, in order to find a counter example, I know this must be satisfied: | x - y | < 5...
  23. E

    Prove trigonometric identity and determine a counterexample

    Homework Statement cos(x-y)cosy-sin(x-y)siny=cosx a.try to prove that the equation is an identity b. determine a counterexample to show that it is not an identity Homework Equations cos(x-y) = cosxcosy+sinxsiny sin(x-y) = sinxcosy-cosxsiny The Attempt at a Solution a.Left side of...
  24. alexmahone

    MHB Is the Harmonic Series a Counterexample to a Convergent Series?

    Give a counterexample to $na_n\to 0,\ a_n\ge 0,\ a_n$ decreasing $\implies\sum a_n$ converges.
  25. F

    Prove or Counterexample problem

    Hi - My first post here and was looking for some help with this problem. Not sure where to start so hope some pointers would get me going/thinking! Q: Suppose that there is a party with n ≥ 2 people and that each person gives presents to one or more people at the party (but no more than one...
  26. N

    Counterexample intersections of 2 compacts is compact ?

    Counterexample "intersections of 2 compacts is compact"? Hello, I'm looking for a counterexample to "If A and B are compact subsets of a topological space X, then A \cap B is compact." It's not for homework. I found one online, but it talked about "double-pointed" things which I didn't...
  27. T

    I can't think of a counterexample to disprove this set theory theorem

    I can't think of a counterexample to disprove this set theory "theorem" Assume F and G are families of sets. IF \cupF \bigcap \cupG = ∅ (disjoint), THEN F \bigcap G are disjoint as well.
  28. S

    Find a counterexample for a false statment about independent events.

    "Construct a sample space to show that the truth of this statement P(A\bigcapB\bigcapC)=P(A)*P(B)*P(C) is not enough for the events A,B,C to be mutually independent. Hint: Try finite sample spaces with equally likely simple events." So, my though is that I need to find a sample space with...
  29. I

    Proving if R1 \ R2 is Transitive or Not

    Homework Statement Suppose R1 and R2 are relations on A. Therefore, R1 \subseteqA X A and R2 \subseteqA X A Homework Equations Let (x, y) and (y, z) \inR1. Then since R1 is transitive, xR1y, and yR1z implies xR1z. Does R1\R2 mean: (x,y)...
  30. A

    Discrete Math- Irrational numbers, proof or counterexample

    Homework Statement Determine if the statement is true or false. Prove those that are true and give a counterexample for those that are false. If r is any rational number and if s is any irrational number, then r/s is irrational. Homework Equations A rational number is equal to the...
  31. J

    Incredibly close to a modular arithmetic proof by minimum counterexample

    As stated in the title, I am trying to prove a statement by minimum counterexample involving modular arithmetic. My problem is producing the contradiction, but I feel so close! (The contradiction is p^m | (1 + p)^{p^{m - 1}} - 1) Homework Statement Let p be an odd prime and let n be a...
  32. M

    Counterexample to uniqueness of identity element?

    (Hopefully, this question falls under analysis. I was unable to match it well with any of the forums.) The proof that the identity element of a binary operation, f: X x X \rightarrow X, is unique is simple and quite convincing: for any e and e' belonging to X, e=f(e,e')=f(e',e)=e'. However...
  33. WannabeNewton

    Counterexample for set identity

    Homework Statement Consider the function f:X \to Y. Suppose that A and B are subsets of X. Decide whether the following statements are necessarily true (I am including just the one I had trouble with): (a) if A\cap B = \emptyset , then f[A]\cap f[B] = \emptyset Homework Equations The...
  34. G

    Looking for Gauss-Bonnet counterexample

    Hello everybody! I was looking for a counterexample to Gauss-Bonnet Theorem, that is, a region R \subset \Sigma (with \Sigma \subset \mathbb{R}^3 surface) such that \partial R isn't union of closed piecewise regular curves and for which the Gauss Bonnet Theorem doesn't holds, i.e. \iint_R{K...
  35. Z

    Is there any counterexample to this ?

    is there any counterexample to this ?? let be the Fourier transform G(s) = \int_{-\infty}^{\infty}dxf(x)exp(isx) with the properties f(x) and D^{2}f(x) are EVEN funnctions of 'x' f(x) > 0 and D^{2}f(x) > 0 on the whole interval (-oo,oo) then G(s) has only REAL roots...
  36. R

    Confused with using Proof by Smallest Counterexample

    When proving by smallest counterexample, you assume an integer k>1 where it is the smallest integer for which statement Sn is false. Then you proceed to prove that Sk-1 implies Sk. Where you deduce a contradiction by which k is true. Can't you prove this directly by assuming Sk-1 is true and...
  37. R

    Can the Product of n Consecutive Positive Integers Be Divisible by n!?

    Homework Statement Using the minimal counterexample technique prove that the product of n consecutive positive integers is always divisible by n! The Attempt at a Solution Suppose that the statement is not true and the product of n consecutive positive integers is not always divisible by...
  38. L

    Proof using counterexample. HELP

    1. Prove by minimum counterexample that for all n>=0, 5/(32n)-4n) 2. Homework Equations : proof by induction? 3. I tried plugging in 0 for n because that would be the minimum counterexample since 5 can't divide 0. If it's not zero it might be 2 because that works as well. I'm not sure...
  39. D

    Testing the Convergence of Series: A Counterexample

    Homework Statement If \sum_{k=1}^{\infty} a_k converges and a_k/b_k \to 0 as k\to \infty, then \sum_{k=1}^{\infty} b_k converges.Homework Equations It is true or false.The Attempt at a Solution I think it is false and here is my counterexample. Let a_k = 0,b_k=\frac{1}{k}. This satisfies our...
  40. V

    Counterexample for a premeasure on a semiring over Q

    Homework Statement Let \alpha(r)=r and let P be the family of intervals [a,b) in \mathbb{Q}. Define \mu_{\alpha}([a,b))=\alpha(b)-\alpha(a). Show by example that \mu_{\alpha} is not countably additive.Homework Equations \mu is countably additive if for any sequence of mutually disjoint subsets...
  41. E

    Delta-System Lemma counterexample

    The \Delta-system lemma states the following: given an infinite cardinal \kappa, let \theta > \kappa be a regular cardinal such that \forall \alpha < \theta \ (|\alpha^{< \kappa}| < \theta); given A such that |A| \geq \theta and \forall x \in A \ (|x| < \kappa), then there is a B \subset A which...
  42. R

    Counterexample: (u + v)^2 ≠ u2 + v2

    Find a counterexample to the statement "For all real numbers u and v, (u + v)^2 is not equal to u2 + v2."
  43. F

    Is There a Counterexample to the Function Composition Property?

    Homework Statement Let f: A\to B. I'm trying to find a function g: B\to C such that g is not 1-1 but g\circ f is. The original assignment (which I've completed) was to prove that for all functions f: A\to B and g: B\to C, if g\circ f is 1-1, then so is f. However, in the process of...
  44. C

    Infinite dimensional counterexample

    Homework Statement Let V be a finite dimensional vector space and let W be a subspace of V. 1. Then V is the direct sum of W and W' where W' denotes the orthogonal complement of W. 2. Also, (W')' = W, i.e the orthogonal complement of the orthgonal complement of W is again W. My...
  45. S

    Is the Inverse of a Continuous Function Always Continuous?

    Homework Statement Let ( X, \tau_x) (Y, \tau_y) topological spaces, (x_n) an inheritance that converges at x \in X, and let f_*:X\rightarrow Y[/itex]. Then, [tex]f[/itex] is continuos, if given (x_n) that converges at [tex]x \in X , then [tex]f((x_n))[/itex] converges at...
  46. P

    Needs a counterexample for homomorphisms

    Homework Statement Let A, B be groups and A' and B' be normal subgroups of A and B respectively. Let f: A --> B be a homomorphism with f(A') being a subgroup of B'. There is a well-defined homomorphism g: A/A' -----> B/B' defined by g: aA' ---> f(a)B' Find an example in which f is...
  47. C

    Counterexample and Proving for Sets A, B, C

    Homework Statement 1. Provide a counterexample to the following conjecture: For sets A, B, C \subseteq U if A is a subset of B but B is not a subset of C, then A is not a subset of C 2. (A\cap B) \cup C = (A \cap (B \cup C)) if and only if C \subseteq A 3. Prove (A - B) - C = (A...
  48. S

    Simple counterexample for claim about integral domains

    So I'm looking for an example of an infinite integral domain with finite characterestic. That is a infinite integral domain such that there is a prime p such that p copies of any element added together is the additive identity. I'm just looking for a simple counterexample. I'm working...
  49. E

    Solving 1.8.5 Part b): Is It a Counterexample?

    Homework Statement Is what I wrote on the left hand margin a counterexample to 1.8.5 part a) ? EDIT: I meant part b) Homework Equations The Attempt at a Solution
  50. S

    Counterexample for an alternate definition for a group

    This is another problem from Herstein. The first problem 12 asks to prove that if G is closed under associative product and (a) there is a e so that for any element a in G we have ae=a (b) for each a in G there is a y(a) so that ay(a)=e then G is a group. Although it took me some time to do...
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