Questions tagged [fake-proofs]
Seemingly flawless arguments are often presented to prove obvious fallacies (such as 0=1). This is the appropriate tag to use when asking "Where is the proof wrong?" about proofs of such obvious fallacies.
1,258
questions
1
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1
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Circumference of an ellipse is π(a + b)?
I know that a formula for an ellipse in terms of elementary functions doesn't exist. But I seem to have found an intuitive false-proof that the circumference of an ellipse with axes $a$ and $b$ equals ...
- 13
4
votes
1
answer
109
views
What is wrong with my argument that every group of order $pq$ is abelian?
Question:
Prove that a group $G$ of order 143 is abelian.
Solution:
$G$ contains subgroups $H$ and $K$ of orders 11 and 13, respectively.
The subgroups $G/H$ and $G/K$ have orders 13 and 11, ...
- 96
0
votes
1
answer
55
views
What's wrong with this integration over the volume of a sphere (from Gauss' Theorem)?
I'm struggling to find my mistake in the following problem:
Let $S$ be a sphere of radius $R$ with center at the origin. Let $f(x,y,z) = 3z^2$. Find the integral of $f$ over the volume of the ...
- 2,790
0
votes
1
answer
39
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Incorrect proof for $e$ idempotent $\Rightarrow$ $eA$ projective
I have seen several proofs for $e$ idempotent $\Rightarrow$ $eA$ projective where $A$ is an algebra. I tried something different and produced a proof without using the fact that $e$ is idempotent (so ...
- 1,640
-1
votes
1
answer
79
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Spot the error in the following "proof" by induction. [duplicate]
The following statement is false, but still, a proof is given. I need to determine the error.
"In any set of $n≥1$ circles in $R^2$, all circles have the same radius".
Base case: If $n=1$, ...
- 1
3
votes
1
answer
73
views
Prime avoidance for graded Noetherian ring with infinite residue field.
This question is related to this question.
Let $R$ be a Noetherian local ring with infinite residue field. Let $\text{gr}_IR$ denote the associated graded ring $\bigoplus_{n=0}^{\infty} I^n/I^{n+1}$. ...
- 443
3
votes
1
answer
97
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I can't find the mistake in my proof
I thought about something in power series and I proved a theorem that I have never seen before:
The theorem:
Let $\begin{align} \sum_{n=1}^{\infty} a_nx^n \end{align}$ be a power series, with a ...
- 347
-2
votes
0
answers
30
views
setting epsilon in sequence limit to the absolute value difference
Sequence is said to have a limit $L$ if:
$$
\forall\epsilon\in\mathbb{R}>0.\exists N\in\mathbb{R}.\forall n\in\mathbb{R}>N.\left| a_{n}-L \right|<\epsilon
$$
since the absolute value is a ...
- 97
1
vote
1
answer
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How can I find the determinant of this $2\times 2$ block matrix directly?
I wish to explicitly compute the determinant of
$$ K = \begin{pmatrix} I & A \\
B & 0 \end{pmatrix} $$
where $A,B$ are $n\times n$ matrices and $I$ is the $n\times n$ identity, using the ...
- 123
0
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0
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45
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Show that $B_1(0)$ is a closed set in the space $C([0,1])$
Let be $(C([0,1]),\Vert \cdot\Vert_{\infty})$ the normed space of continuous functions, equipped with the supremum norm $\Vert\cdot\Vert=\sup\limits_{x\in[0,1]}|f(x)|$.
Show that $B_1(0):=\{f\in C([0,...
- 4,278
4
votes
0
answers
118
views
Symmetries and KdV
I would like to find the value of $\alpha$ such that
$$
V=\frac{\partial}{\partial u}+\alpha t \frac{\partial}{\partial x}
$$
generates a Lie symmetry of the KdV equation.
I found that the generated ...
- 2,207
-1
votes
2
answers
103
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Prove that a ring of order $6$ can never be an integral domain. [duplicate]
Prove that a ring of order $6$ can never be an integral domain.
My solution:
Let $R$ be a ring of order $6$ which is an integral domain. This means, that $1+1\neq 0\in R$ and we note that, $(1+1)(1+1+...
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votes
4
answers
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views
What's wrong with this limit solution?
I was solving this limit:
$$
\lim_{x\rightarrow +\infty} \left( \sqrt{x^2+x+1}-x \right)
$$
And here is my solution:
$$
\lim_{x\rightarrow +\infty} \left( \sqrt{x^2+x+1}-x \right) =\lim_{x\rightarrow ...
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2
votes
1
answer
270
views
where is the error in fake proof that a gaussian has 0 variance
Consider the exponential family $$p_\theta(dy) = h(y)\exp(\langle \theta,T(y)\rangle - \Phi(\theta))\mu(dy)$$ with $\theta \in \mathbb R^d$ and the partition function $\Phi(\theta)<\infty$ for all $...
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1
vote
1
answer
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Cyclic group of order $4$
I'm studying Artin's Algebra book and got stuck in the following problem:
(a) Let $G$ be a group of order $4$. Prove that every element of G has order $1, 2$ or $4$.
My (incorrect) reasoning was: if ...
0
votes
0
answers
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views
In differentiation can you prove the product rule using the sum rule by breaking apart the product
for example if you let
$f(x)=g(x)h(x)$
then $f(x)$ can be written as $f(x)=g(x)+g(x)+g(x)...$ appearing $h(x)$ amount of times
therefore using the sum rule, $f'(x)$ can be written as
$f'(x)=g'(x)+g'(...
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votes
0
answers
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$P(T\in dt,T\leq C|min(T,C)\geq t)=P(T\in dt|T\geq t)$ for $T,C$ independent
Let $T,C$ be two real valued independent random variables. I want to see the following equality, where $dt$ denotes interval $[t,t+\delta)$ for any $\delta>0$
$P(T\in dt|T\geq t)=P(T\in dt,T\leq C| ...
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1
vote
2
answers
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Where is the error in this proof using induction?
We want to prove that $\sum_{i=1}^{n} x_{i}^2 = \sum_{i=1}^{n} x_{i}t_{i}$ only if $t_i=x_i$. We use a proof by induction.
We check the first case: $\sum_{i=1}^{1} x_{i}^2 = \sum_{i=1}^{1} x_{i}t_{i}$ ...
- 1,152
0
votes
1
answer
114
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square root of x equals -1
I read that $\sqrt{x} = -1$ has no solution because after we square both sides we get $x = 1,$ which isn't a correct solution. But doesn't writing $-1$ as $i^2$ give the solution $x = i^4$ ?
$$\sqrt{x}...
- 21
1
vote
2
answers
174
views
Let $f :\mathbb R \rightarrow\mathbb R$ be continuous function . Then prove that $(0,1)$ cannot be the image of $(0,1]$?
Let $f :\mathbb R \rightarrow\mathbb R$ be continuous function. Then prove that $(0,1)$ cannot be the image of $(0,1]$ i.e $f((0,1])\neq (0,1)$ ?
My solution:
Let us assume the contrary that, ...
- 2,356
-4
votes
1
answer
88
views
False proof that $\tfrac{0}{0} = 2.5$ [closed]
I was just playing around with the quadratic formula and wondered what if the a coefficient is $0$, so I took a normal equation like $2x-5=0$ and slapped a $0x^2$ term in front of it to get a ...
1
vote
1
answer
51
views
Not getting where is the mistake in the solution of a problem in Linear Algebra.
The set of all $n×n$ matrices having trace equal to zero is a subspace $W$
of $M_{n×n}(F).$ Find a basis for $W.$ What is the dimension of $W?$
To solve this problem, I first noted that the standard ...
- 2,356
0
votes
0
answers
71
views
What's wrong with the following limit evaluation?
\begin{align*}
\lim_{x \to \infty} \ln{x}&=\lim_{x \to \infty}\frac{\ln{x}}{x}\cdot x\\
&=\lim_{x \to \infty}\frac{\ln{x}}{x}\cdot \lim_{x \to \infty} x\\
&=\lim_{x \to \infty}\frac{(\ln{x}...
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1
vote
3
answers
229
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Let $S =\{u_1, u_2,...,u_n\}$ be a finite set of vectors. Prove that $S$ is linearly dependent iff $u_1 = 0$ or $u_{k+1} ∈ span(\{u_1, u_2,...,u_k\})$
Let $S =\{u_1, u_2,...,u_n\}$ be a finite set of vectors. Prove that $S$ is linearly dependent if and only if $u_1 = 0$ or $u_{k+1} ∈ \text{span} (\{u_1, u_2,...,u_k\})$ for some $k\space (1 ≤ k<n)....
- 2,356
3
votes
0
answers
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views
Is this proof circular or is it valid?
I am playing with the chernoff bounds and derived a bound from its additive version. The method I used seems a little suspicious and circular in logic, but I cant figure out what the mistake is.
...
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1
vote
1
answer
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What's wrong with my simplification of these exponentials?
The question states:
Simplify $2^{\sqrt2}$
I tried doing:
$$2^{\sqrt2}=2^{(2^{\frac{1}{2}})}$$
$$=(2^2)^{\frac{1}{2}}$$
$$=4^{\frac{1}{2}}$$
$$=2$$
$$=2^1$$
Clearly this is ridiculous as I've just &...
- 461
3
votes
0
answers
148
views
Flaw in proof about an exact quadrature method
Consider the quadrature method
$$
\int_{-1}^1f(x)dx \approx \sum_{k=0}^N w_kf(x_k),
$$
where $x_0=-1, x_N=1$, and $x_1,\ldots,x_{N-1}$ are the roots of the derivative of the degree-$N$ Legendre ...
- 1,016
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votes
1
answer
77
views
(Fake proof) Counterclockwise contour integral of identity function around unit circle is $-2\pi i$
First, the result is obviously false by Cauchy's integral formula, given that the identity function is one of the simplest analytic functions and has no singularities. So the contour integral is zero.
...
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votes
2
answers
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views
Find the values of $p$ and $q$ such that $\lim_{x\to 0}\frac{x(1-p\cos x)+q\sin x}{x^3}=\frac 13.$ Assume, that L' Hospital's rule is applicable.
Find the values of $p$ and $q$ such that $\lim_{x\to 0}\frac{x(1-p\cos x)+q\sin x}{x^3}=\frac 13.$ Assume, that L' Hospital's rule is applicable.
I tried solving the problem as follows:
Given, $\lim_{...
- 2,356
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vote
0
answers
60
views
Show Tree Proof is Incorrect
It seems the part that says every vertex in H has degree at least two is not necessarily true. The paths could converge prior to seeing vertex v, thus resulting in at least one vertex having degree 1. ...
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1
vote
1
answer
107
views
Is this proof of Darboux's Theorem valid?
Let $f:\Bbb [a,b]\to \Bbb R$ satisfy the following :
(i) $f(x)$ is derivable in $[a,b]$
(ii) $f'(a)=\alpha\neq f'(b)=\beta$ and
(iii) $\exists \gamma\in (\alpha,\beta)$
then, there exists at least one ...
- 2,356
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votes
1
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views
Rudin's proof about closed subsets of compact sets
The theorem, and proof presented in Rudin is:
Theorem: closed subsets of compact sets are compact
Proof: Suppose $F \subset K \subset X$, $F$ is closed (relative to $X$), and $K$ is compact.
Let $\\{...
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votes
0
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48
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Won't signed measures upset Riemann?
I can't seem to wrap my head around signed measures.
If $\mu$ is a signed measure on a measurable space $(X, \Sigma)$, then there'll be sets of both positive and negative measure. Let $E_1, E_2, \...
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votes
1
answer
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u-substitution shows 0=1
This question/observation is inspired by the integral:
$$\int_0^{\sqrt{\pi}}x\sin(x^2)\cos(x^2)dx$$
The $u$-substitution $u=\sin(x^2)$ yields $du=2x\cos(x^2)dx$ and
$$\int_0^{\sqrt{\pi}}x\sin(x^2)\cos(...
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votes
1
answer
48
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Problem when factoring operators
So I was trying to find a general solution to the second-order linear differential equation: $$\frac{d^2y}{dx^2}+p(x)y=0$$I had read about factoring operators as if they were polynomials. So, I wrote ...
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votes
2
answers
120
views
Why is this algebraic proof incorrect? [closed]
This is a simple proof of the statement:
\begin{equation}x^y=y^x\quad\quad(1)
\end{equation}
In which i use the identity
$$a=\log_b(a^{a\log_a(b)})$$
To show the validity of the identity, we use the ...
1
vote
0
answers
191
views
Why series expansion doesn’t work
The Euler Maclaurin Formula is $$\sum_{k=0}^x f(k)=\int_0^xf(t)dt+\frac{f(x)+f(0)}2+R_1\tag{1}$$Where $R_1$ is the remainder term defined as (according to Wikipedia)$$R_p=\frac{(-1)^{p+1}}{p!}\int_0^...
- 5,052
0
votes
1
answer
45
views
Is the statement about continuity of a linear function based on the closedness of its kernel always true?
I would like to answer this:
Let $f : V \rightarrow \mathbb{R}$ be a linear function with $V$ being a normed vector space (NVS). Prove that $f$ is continuous if and only if $\text{ker} \, f := f^{-1}(...
- 2,207
0
votes
1
answer
119
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A proof that $\int_{-1}^1\frac1xdx=0$
I was inspired by BlackPenRedPen's video about why $$\int_{-1}^1\frac1xdx$$ doesn't exist. However, I was able to prove the following claim which gives the above integral as $0$. I want to know what ...
- 5,052
1
vote
1
answer
46
views
Finding a recursive formula for $r$ permute $n$
I'm learning about permutations, and have been playing around with them trying to find a recursive definition of $P(n+1,r)$ in terms of $P(n,r)$, where $P(n,r)$ represents the number of $r$ ...
- 402
1
vote
2
answers
77
views
Question on a proof that a space of linear map between normed spaces is a Banach space when the arrival space is a Banach space
I have a question concerning the proof of the result stated in the title.
To prove this theorem, we consider $(f_n)_{n\in\mathbb{N}}$ a Cauchy sequence in $L(E,F)$ :
$$
\forall\varepsilon>0, \...
- 1,012
0
votes
1
answer
73
views
if $f[X]=f[Y]$ then $X=Y$???
I'm pretty sure this is false, but this proof I came up with seems to be valid.
Proof.
If $f[X]=f[Y]$, then for any $x\in X$, $f(x)\in f[X]=f[Y]$, now $f(x)\in f[Y]=\{f(x)|x\in Y\}$, so $x\in Y$; ...
- 177
0
votes
0
answers
14
views
Positive quadratic function in $\mathbb{R}$ for all real numbers requires a negative determinant
The following simple exercise appears in the book "Undergraduate Topology" by Robert H. Kasriel (Dover publication 2009). I know you can find proofs in any elementary algebra book, yet I ...
- 86
1
vote
1
answer
85
views
What is wrong with the following proof of the Second Fundamental Theorem of Integral Calculus?
An exercise in a textbook I am using to study integral calculus posits the following query: what is wrong with the following proof of the Second Fundamental Theorem of Integral Calculus? (The theorem ...
- 1,387
0
votes
0
answers
28
views
If $A_i$ and $B_i$ are subset of $(X_i,*_i)$ then does the equality $\prod_{i∈I}A_i⋆\prod_{i∈I}B_i=\prod_{i∈I}(A_i⋆_i B_i)$ holds?
Let be $\mathfrak X$ a indexed collection over a set $I$: so it is a well know result that if $*_i$ is for each $i$ in $I$ an operation on $X_i$ in $\mathfrak X$ then the equality
$$
(a*b)(i):=a(i)*_i ...
- 4,708
2
votes
2
answers
99
views
Prove that the equalities $A(B\cap C)=AB\cap AC$ and $(A\cap B)C=AC\cap BC$ do not generally hold.
If $*$ is a binary operation on a set $X$ then it is custom to define
$$
\tag{1}\label{1}A\star B:=\{x\in X:x=a*b\text{ with } (a,b)\in A\times B\}
$$
for any $A,B\in\mathcal P(X)$.
So my algebra text ...
- 4,708
1
vote
1
answer
44
views
Why are the probable candidates getting eliminated with this line of reasoning?
If $p$ is a prime prove that, $(p-1)!+1$ is a power of $p$ iff $p=2,3,5$.
I proceeded by assuming that $(p-1)!+1$ is a power of $p.$ So, $(p-1)!+1=p^k,$ $k\in\Bbb Z$. This implies $p^k-1=(p-1)!$ and ...
- 2,356
1
vote
1
answer
123
views
Pdf of the radius of a 2D Gaussian for a fixed angle
The 2D Gaussian distribution is known for its rotational symmetry (as shown in this 3blue1brown video), and it's also known that the radius $R = \sqrt{X^2 + Y^2}$ in polar coordinates follows a ...
- 1,177
0
votes
0
answers
82
views
$i^3=1$, this must be false but [duplicate]
Ok this is quite dumb and absurd but it goes like this:
$i^3 = (i^4)^{3/4}= 1^{3/4} = 1$
I am not very sure what the mistake is, maybe if this was true we contradict $(i^4)^{3/4} \neq (i^{3/4})^3$
- 9
0
votes
3
answers
131
views
Using Euler's identity [duplicate]
Using Euler's identity, I was able to prove that $i = 0$, and I can't find where I went wrong.
Euler's identity, I recall being the following:
$$e^{i \pi} = -1$$
From there, with a few changes like so:...
- 11