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.
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##...
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...
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...
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...
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) =...
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...
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...
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...
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...
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...
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.
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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.
"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...
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)...
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...
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...
(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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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
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...