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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.

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Can $\text{rank} (T) + \text{nullity} (T) = \dim V$ be proven with this simple argument?

I am helping one of my friends with linear algebra and gave him this theorem to prove as an exercise: Theorem . Let $V$ and $W$ be vector spaces over the field $F$ and let $T$ be a linear ...
Mathematics enjoyer's user avatar
1 vote
2 answers
88 views

Proof about adjoint matrix

Prove: If $A_{n\times n}$ is not invertible then $Adjoint(A)$ is not invertible also. I have made the following: By contradiction. Suppose that $Adj(A)^{-1}$ exists Then write $Adj(A)*B=I$ \begin{...
DEMB's user avatar
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2 votes
2 answers
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Why this simple volume problem in Multivariate Calculus seems to have an anomaly?

Find the volume of the solid inside the cylinder $x^2+y^2-2ay = 0$ and between the plane $z = 0$ and the cone $x^2+y^2 = z^2$. I tried solving this problem as follows: Equation of the cylinder $x^2+(y-...
Thomas Finley's user avatar
2 votes
2 answers
49 views

False proof $(Av)^T=v^TA.$

Given $u^Tv = u\cdot v=v\cdot u=v^Tu,$ I tried to prove the well-known $(Av)^T=v^TA^T$ so I did the following $$(Av)^Tv=v^T(Av)=(v^TA)v$$ and got $(Av)^T=v^TA$ instead. Can someone explain what went ...
user7777777's user avatar
1 vote
1 answer
46 views

Validity of an inductive step for a problem of probability

Consider the next problem Problem. Let $A_1,...,A_n$ be events. We assume that at least one event occurs, but never more than three. We also assume that the probability of having at least two events ...
RataMágica's user avatar
3 votes
3 answers
154 views

What's wrong with this "proof"? Induction Fibonacci Big-Oh

I can't find out what is wrong in this (computer science) induction exercise. Maybe I'm not cut out for this stuff. Consider the following "proof" that the Fibonacci function, $F(n)$, ...
Science Guy's user avatar
6 votes
1 answer
668 views

Mistake in Proof "Every unique factorization domain is a principal ideal domain"

While doing my Algebra HW, I "proved" that every unique factorization domain (UFD) is a principal ideal domain (PID). I know that this is not true, however I fail to see where exactly is ...
DeeJeiK's user avatar
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5 votes
1 answer
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Show that $\int_0^1\frac{1}{(x+1)(x+2)\sqrt {x(1-x)}}$ is convergent.

Show that $\int_0^1\frac{1}{(x+1)(x+2)\sqrt {x(1-x)}} \,dx$ is convergent. The points $0$ and $1$ are the only points of infinite discontinuities of $\frac{1}{(x+1)(x+2)\sqrt {x(1-x)}}.$ The integral ...
Thomas Finley's user avatar
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0 answers
33 views

Where is the flaw in this argument about line segment length? [duplicate]

Examine the line segment on the Cartesian plane between $(0, 0)$ and $(1, 1)$. This line segment can be formulated using set notation as $L = \{x=y, 0 \leq x \leq 1\}$. This line connects $(0, 0)$ to $...
Ryan Folks's user avatar
6 votes
3 answers
789 views

Where is the mistake in the argument in favor of the (erroneous) claim "every Dedekind cut is a rational cut"?

A cut is a set $C$ such that: (a) $C\subseteq \mathbb Q $ (b) $C \neq \emptyset $ (c) $C \neq \mathbb {Q} $ (d) for all $a, c \in \mathbb Q $ , if $c\in C$ and $a\lt c$ , then $a\in C $ (e) for all $c\...
Vince Vickler's user avatar
2 votes
2 answers
98 views

A "proof" of negative variance

In trying to solve the problem below, I got a negative variance. What is the error in this "false proof" of negative variance? And why does it produce the correct variance times $-1$? Let $...
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1 vote
1 answer
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Contradiction: If and only If, an apparent "shortcut". What is the mistake?

I noticed something while I was proving an if and only if statement, and it seems to provide a "shortcut" that I haven't seen before. Say I want to prove $A \Leftrightarrow B$. For $A \...
rossignol's user avatar
1 vote
4 answers
129 views

How does $|a+bi| = \sqrt{a^2+b^2}$? If I use algebra and break it down I just get to $1=-1$.

I have gotten to a chapter in my pre-calculus textbook where it mentions this equation: $|a+bi| = \sqrt{a^2+b^2}$ It doesn't really explain why that equation is true, and it just sort of moves on ...
Darkturtle999's user avatar
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Paradox $1<1$ using complex exponentiation [duplicate]

\begin{align*} 1&>e^{-4\pi^2} \\ &= e^{(2\pi i)(2\pi i)} \\ &= (e^{2\pi i})^{2\pi i} \tag{$\ast$} \\ &= 1^{2\pi i}\\ &= 1 \end{align*} The problem with this fake proof is that ...
J P's user avatar
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6 votes
1 answer
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A false "proof" that record setting events are dependent

Let $\{X_i\}$ be a sequence of i.i.d continuous RV. Call $i$ record-setting if $$X_i > \max_{1 \leq j < i} X_j.$$ It is well-established on math.SE and elsewhere that the events "$X_n$ is ...
SRobertJames's user avatar
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-1 votes
1 answer
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$z^p =(z^n)^{p/n} =1$? where z is $n^{th}$ root of unity and $p \in \mathbb R$ [duplicate]

We know that there are $n$ solutions for $n^{th}$ root of unity, $z^n = 1$ : $1, z, z^2, z^3, ......z^{n-1}$ where, $z = e^{i2\pi/n}$ Now, $$z^{n+1} = z$$ right? Since, $$z^{n+1} = z^n.z = 1.z=z $$$$.:...
Ishant Dumane's user avatar
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0 answers
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Proving $e^{i \theta} =1$ [duplicate]

Let $\theta \in (-\pi,\pi) $ we have that $$ e^{i\theta} = \left(e^{i\theta}\right)^\frac{2\pi}{2\pi} = (e^{2\pi i})^{\frac{\theta}{2\pi}} $$ so as $e^{2\pi i}=1$ we have that $$ e^{i\theta} = (1)^{\...
baristocrona's user avatar
1 vote
4 answers
209 views

Density of $\mathbb{Q}$ in $\mathbb{R}$ seemingly contradictory to infinite set cardinality [duplicate]

In my real analysis module we proved that both $\mathbb{Q}$ and $\mathbb{R \setminus Q}$ are dense in $\mathbb{R}$, meaning that between any two real numbers there exists a rational and an irrational ...
Darya's user avatar
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3 answers
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What is wrong about this seemingly simple false proof? [duplicate]

It came accros my mind when doing an exercise in calculus. Consider the following inequalities: $1\leq 2\leq 5$ $1\leq 3\leq 5$ Adding the equaions up is totally fine. But subtracting them, we get: $0\...
natitati's user avatar
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0 votes
1 answer
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$f$ is an entire function, then $f(\mathbb{C})$ is closed

While trying to prove "$f$ is an entire function such that $|f(z)| \to \infty, as |z|\to \infty$, then $f(\mathbb{C}) $ is closed" I accidentally showed that "$f$ is an entire function, ...
Praveen Kumaran P's user avatar
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1 answer
90 views

Cantor's diagonal argument for proving the completeness of $L^\infty$ space, check work

I wrote up the following proof for proving $L^\infty$ is complete. Please check if I made any mistakes, thank you! There is a common/standard treatment for establishing the limit of a Cauchy sequence ...
William Chuang's user avatar
3 votes
0 answers
134 views

Fermat's (hypothetical) erroneous proof

Until Wiles' proof of Fermat's last theorem all proposed proofs have been erroneous. It is not known which proof Fermat himself had in mind - but it is assumed that it was erroneous, too. Have there ...
Hans-Peter Stricker's user avatar
0 votes
0 answers
69 views

Understanding the logic behind my "proof" of a false statement

I was working through these questions: (c) Prove that f(C1 ∩ C2) ⊆ f(C1) ∩ f(C2). (d) Give a counterexample to the opposite inclusion in (c). The opposite of c. (...
sharkleberryfin's user avatar
3 votes
1 answer
117 views

Confusion about projective resolution: group cohomology

I am totally confused about a detail in computing group cohomology. Let $G$ be a group, $A$ a $\mathbb{Z}[G]$-module. $H^i(G, -)$ is the $i$th right derived functor of $\operatorname{Hom}_{\mathbb{Z}[...
Emory Sun's user avatar
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-3 votes
1 answer
67 views

Is my proof wrong for: $n^k \in O(2^n) \text{ for any constant } k$ [closed]

I tried to prove this the following way, but not sure if this is correct? $n^k \leq c \cdot 2^n$ $\log_2(n^k) \leq \log_2(c \cdot 2^n)$ $\log_2(n^k) \leq \log_2(c) + \log_2(2^n)$ $\log_2(n^k) \leq \...
Liz's user avatar
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0 votes
0 answers
38 views

On a proof of the converse supporting hyperplane theorem

In connection with reading a proof of a theorem in a real-analysis textbook, I've stumbled upon the following theorem. Theorem. Suppose $C$ is closed with non-empty interior and has a supporting ...
psie's user avatar
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0 votes
3 answers
131 views

$\displaystyle\lim_{n\to\infty} \frac{\sin \frac{1}{n}}{\frac{1}{n}} = 0$, where is my mistake?

$\displaystyle\lim_{n\to\infty} \frac{\sin \frac{1}{n}}{\frac{1}{n}} = 1$, but when I evaluate it as follows I get $0$ as the result: \begin{align} \lim_{n\to\infty} \frac{\sin \frac{1}{n}}{\frac{1}{n}...
VincentSchaerl's user avatar
1 vote
1 answer
93 views

Where have I gone wrong in computing the integral closure of $k[X, Y, Z]/(Y^3 + Y^2X^2 + YX^2 + X^3Z)$?

I was trying to follow the strategy in this answer to compute the integral closure of the domain $$R := k[X, Y, Z]/(Y^3 + Y^2X^2 + YX^2 + X^3Z)$$ where $k$ is a field. During the computation, I ...
Anakhand's user avatar
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1 vote
1 answer
57 views

A faulty "proof" regarding exactness of derived functors

I've been trying to wrap my head around derived functors and have come upon the following chain of arguments, which seems to yield the unreasonabel conclusion that all left derived functors are exact. ...
Jeppe Obel's user avatar
-1 votes
2 answers
91 views

sin/cos = cos/sin

To begin with, let's note the following: $$ \tan = \frac{\sin}{\cos} \\ \sin = \tan \cdot \cos \\ \cos = \frac{\sin}{\tan} $$ If we replace one of the functions in $\frac{\cos}{\sin}$ according to the ...
Damian Czapiewski's user avatar
0 votes
0 answers
29 views

Where Error in evaluate summmation :$\sum_{k=0}^{m}\binom{n-k}{m-k}$ [duplicate]

Where Error in evaluate summmation :$$\sum_{k=0}^{m}\binom{n-k}{m-k}$$ I know that : $$\sum_{k=0}^{m}\binom{n-k}{m-k}=\binom{n+1}{m}$$ But I does't know error My proof this : we have : $$\binom{n-k}{...
Mostafa's user avatar
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0 votes
1 answer
35 views

Getting different answers to a problem when using different methods. Cannot find a problem with either.

The question I'm solving is: A student jogs from home to the store. She runs at 1 m/s for 1/3 of the time and 1.5 m/s for the remaining time. On her way back home she runs at 1 m/s for half the ...
TheCosmicAspect's user avatar
1 vote
0 answers
134 views

What goes wrong in this "proof" that $ 2 \pi i = 0 $?

My textbook on complex numbers showed this example: $$ \ln(-\sqrt e) = \ln(-1) + \ln(\sqrt e) = \ln(e^{\pi i}) + \ln(e^{\frac 12}) = \tfrac 12 + \pi i $$ Now, using the same logic, I got this: $$ \...
Jeroen van Rensen's user avatar
2 votes
3 answers
160 views

Confusion about proof statement by implication

I have a question regarding proving "if p is true then q is true". One way, is to show p $\implies$ q is a true statement, so then if p is true, then q is true. The other way is to prove it ...
Aria's user avatar
  • 21
1 vote
1 answer
94 views

Find the flaw in this mapping between the naturals and reals

I was studying Cantor's diagonal argument etc. I was testing the ideas and I thought of the following mapping between the naturals and the reals and I need some help to find the flaw in it. For ...
Antonis Karvelas's user avatar
4 votes
2 answers
146 views

A suggested proof of the uncountability (sic) of rational numbers

[Note: I initially made the mistake of posting this over at MathOverflow, where it was promptly closed. There are however some comments there. Link. ] At MathOverflow, I found a very interesting ...
Anders H's user avatar
2 votes
3 answers
144 views

Is it true that if $a_n \to a$ and $\sum b_n \to b$ then $\sum a_n b_n \to ab$

I was trying to prove Abel's Test for convergence and noticed that Since $\sum\limits_{k=1}^\infty b_k$ converges ,then for all $\varepsilon>0$ $\exists N_1\in \mathbb{N}$ such that for all $n,m \...
Mathematics enjoyer's user avatar
0 votes
2 answers
132 views

Where does my short proof of the 4-colour theorem go amiss?

The proof is essentially a generalisation of the $5$-Colour Theorem: Proof. Consider the set $S$ of planar graphs which are not $4$-colourable. Choose the graph $G \in S$ that has the least number of ...
David's user avatar
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3 votes
1 answer
180 views

Fake proof that $\lim \limits_{x \to 0} \frac{\sin(3x)}{x^3}=\frac{-9}{2}$

In the book Mathematical Fallacies, Flaws, and Flimflam, the following fake proof is provided: The question starts with using trigonometric identities to turn $\sin(3x)\to3\sin(x)-4\sin^3(x)$. Then ...
MPK1962's user avatar
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1 vote
1 answer
173 views

Let $V$ be a vector space. Determine all linear transformations $T: V → V$ such that $T = T^2.$

Let $V$ be a vector space. Determine all linear transformations $T: V → V$ such that $T = T^2.$ My solution goes like this: Let $T:V\to V$ be such that $T^2=T.$ Let $F(T)=\{x\in V: T(x)=x\}.$ Claim: $...
Thomas Finley's user avatar
0 votes
1 answer
102 views

Why my proof that $\lim_{x\rightarrow\infty}\frac{x!}{x^x}=1$ is wrong

It is well known that $$\lim_{x\rightarrow\infty}\frac{x!}{x^x}=0$$But according to the definition of the Stirling numbers of the first kind, the falling factorial could be written as $$(x)_n=\sum_{k=...
Kamal Saleh's user avatar
  • 6,549
1 vote
1 answer
126 views

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 ...
aether's user avatar
  • 17
4 votes
1 answer
141 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, ...
Dhawal Patil's user avatar
0 votes
1 answer
89 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 ...
SRobertJames's user avatar
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0 votes
1 answer
48 views

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 ...
kubo's user avatar
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-1 votes
1 answer
89 views

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 $n=1$, ...
lisa's user avatar
  • 1
3 votes
1 answer
103 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 $\bigoplus_{n=0}^{\infty} I^n/I^{n+1}$. ...
Display name's user avatar
3 votes
1 answer
121 views

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 ...
Chess player's user avatar
2 votes
1 answer
78 views

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 ...
RDL's user avatar
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0 votes
0 answers
53 views

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,...
Philipp's user avatar
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