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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.

7
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2answers
95 views

What's wrong with this proof that $0 = 1$?

Let $$f_n(x)=\frac{1}{\sqrt{\pi n}}e^{-x^2/n}.$$ Note that $f_n(x)\to 0$ uniformly as $n\to\infty$. [Proof: $0\leq f_n(x)\leq\frac{1}{\sqrt{\pi n}}$; given any $\epsilon > 0$, let $M=\left\lceil\...
0
votes
2answers
61 views

Irritating “proof” of the Collatz Conjecture

I recently stumbled across this self-proclaimed proof of the Collatz Conjecture. It seems very irritating to me that this very hard conjecture is supposedly proven by using very basic counting ...
0
votes
4answers
35 views

What is the intersection of an event with itself?

The probability of the intersection of two event is: $P(A \cap B) = P(A)P(B)$ If the two events are the same, i.e $P(A \cap A) = P(A)P(A) = P(A)^2$ However, the logic tells us that the probability ...
0
votes
2answers
47 views

Some matrices are real numbers?

Suppose $a\in \mathbb{R}$, $A$ is a matrix, and $I$ is the identity matrix. Then, $$ I a = I a \\ A = Ia\\ IA = IIa \\ IA=Ia\\ \implies A=a $$
4
votes
2answers
210 views

What is wrong with this false proof of $\pi=0$?

Consider the integral $$I=\int_{-1}^{1}\frac{1}{x^2+1}\mathrm{d}x$$ Now, from the standard integral results we know, $$\int\frac{1}{x^2+1}\mathrm{d}x=\arctan(x)+c$$ So, $$\int_{-1}^{1}\frac{1}{x^2+1}\...
0
votes
1answer
37 views

And in this situation can math not work?

Imagine you need to find an unknown like the following: using a Statement A you find a statement B (for example an equality) that let you find that unknown,but when you have statement B you dosen't ...
0
votes
3answers
35 views

Problem with derivative of x^n

I have a problem regarding the proof of the derivative of x^n using first principles. Here's my proof. D is for delta y = x^n y + Dy =(x+Dx) ^n So Dy = (x+Dx) ^n - x^n We can factor this as (x ...
0
votes
1answer
50 views

False Proof of $0^0=1$ [duplicate]

I saw the following argument on Quora that provides which should actually be a false proof since $0^0$ is not defined to be $1$. $(a+b)^{n}=\sum_{k=0}^{n}\binom n{ k}a^kb^{n-k}$. Put $a=0, b=1, n=...
0
votes
0answers
29 views

Generalization and Proof of Littlewood's Conjecture

Does anyone know of the validity of this "Proof" of the Generalization and Proof of Littlewood's Conjecture ? Do you know this Kaveh Mozafari?
8
votes
1answer
74 views

Why isn't $e^n$ equal to 1?

We know $e^{2\pi i} = 1$, and that $(x^m)^n = x^{mn}$. This way, we can rewrite $e^{n}$ as some version of $(e^{2\pi i})^{\frac{n}{2\pi i}}$ for most n (right?). But if this is true, then why isn't ...
0
votes
3answers
81 views

What is the pitfall in the inductive “proof” of P(x):= x does not succeed 1?

The following statement of Peano's axioms appears in Fundamentals of Mathematics, Volume 1 Foundations of Mathematics: The Real Number System and Algebra; Edited by H. Behnke, F. Bachmann, K. Fladt, W....
0
votes
0answers
46 views

false proof of 2 =1, need confirmation [duplicate]

n^2 = n + n + n ... (n times) derivative of n^2 = derivative of n + n + n + n .. (n times) thus, 2n = 1 + 1 + ... (n times) thus, 2n = n and 2 = 1 now i know that the mistake lies in taking the ...
-2
votes
2answers
31 views

How is the “Push-Forward Measure” a measure?

I just watched a video on Measure Theory and am quite sure I had some misunderstanding as to how a measurable map works. The video asserted that a push-forward measure is a proper measure, so for some ...
7
votes
5answers
845 views

Why we can't differentiate both sides of a polynomial equation? [duplicate]

Suppose we had the equation below and we are going to differentiate it both sides: \begin{align} &2x^2-x=1\\ &4x-1=0\\ &4=0 \end{align} This problem doesn't seems to happens with other ...
-3
votes
1answer
76 views

What is wrong with this proof of $i = 0$?

This is a proof I made a year ago and at that time, I didn't see any problems with it. Could anyone point out what is wrong here? Consider the following expresion: $(-1)^{(4n+3)/2}$, where $n \in \...
0
votes
1answer
25 views

q-derivative of binomial

Problem Find the q-derivative of $(a-x)_q^n$ for $n \ge 1$. Answer This is actually equation (3.11) of Kac and Cheung's Quantum Calculus $$D_q(a-x)_q^n=-[n](a-qx)_q^{n-1}.\tag{3.11}$$ My question ...
4
votes
1answer
36 views

In showing that if $V=X \oplus Y$, then $X \cap Y = \{0\}$, is writing $v=0+v$ and $v=v+0$ for $v \in X \cap Y$ sufficient?

$V=X \oplus Y$ if every vector in $V$ can be written uniquely as $v=x+y$ for $x \in X$ and $y \in Y$. Suppose $V=X \oplus Y$. We want to show $X \cap Y =\{0\}$. Suppose not. Let $v \in X \cap Y$. ...
-3
votes
2answers
96 views

$\iota \equiv \pm 3, \pmod{10}$ [closed]

I was reading up on the properties modulo function, when I saw the property: $$-a \equiv (10-a) \space \pmod{10}$$ Which means $$-1 \equiv (10-1) \equiv 9 \space \pmod{10}$$ Now: $$\iota = \sqrt{-...
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votes
3answers
76 views

What went wrong in proving $i=1$ [duplicate]

I started with $$x=(-16)^{\frac{1}{2}}$$ $$x=(-16)^{\frac{2}{4}}$$ Since $$(a^m)^n=a^{mn}$$ we have: $$x=((-16)^2)^{\frac{1}{4}}$$ $$x=((16^2)^{\frac{1}{4}}$$ $$x=\sqrt{16}=4$$ Hence $$(-16)^{\...
0
votes
0answers
49 views

What's wrong in this proof of 1=2? [duplicate]

Here is the proof: let a = b a² = ab a²-b² = ab-b² (a-b)(a+b) = b(a-b) a+b = b 2b = b 2 = 1 Now, of course Problem must lie on the line ...
-2
votes
4answers
88 views

Finding the mistake of my fake proof where $\pm-2=-2$ [closed]

I am wondering what has gone wrong in the following: $4=4 \iff \sqrt{4}=\sqrt{4}\iff\pm 2= \sqrt{4}\iff \sqrt{4}=2\;\text{and}\;\sqrt{4}=-2\iff \pm2=2$ (from substituting back in and obviously ...
9
votes
2answers
209 views

Show that the following sequence converges. Please Critique my proof.

The problem is as follows: Let $\{a_n\}$ be a sequence of nonnegative numbers such that $$ a_{n+1}\leq a_n+\frac{(-1)^n}{n}. $$ Show that $a_n$ converges. My (wrong) proof: Notice that $$ |...
0
votes
1answer
32 views

Two Tests for Exactness

Suppose we have a differential equation of the form $M(x,\ y)\ dx + N(x,\ y)\ dy = 0$, where $\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \neq 0$. It is thus inexact, and absent ...
2
votes
1answer
62 views

Why is independent condition needed in $Pr(X=Y)=0$

I know that if $X,Y $ are independent random variables then $Pr(X=Y)=0$. It kind of make sense to me to ask for independence since $Pr(X=X)=1$. I know the proof of this when $ X,Y$ are discrete but ...
1
vote
2answers
31 views

Show $x_1,…,x_n>0\wedge x_1\cdot…\cdot x_n=1\Rightarrow \sum x_n \geq n$ [duplicate]

Induction does not work here but why not? Case n=1 then $x_1=1$ Inductionstep $x_1.....x_n+1=1\rightarrow x_1...x_n=\frac{1}{x_{n+1}}\overset{IH}{\Rightarrow}x_{n+1}=1$ This must be wrong ...
0
votes
2answers
39 views

Why do we need the commutative axiom of multiplication?

I think I can prove that $ab = ba$ by other axioms. Am I wrong? Why? Edit: proof updated with notes showing where I actually used the commutative property without noticing. Proof. For any numbers ...
4
votes
1answer
84 views

Why does weak-$L^2$ convergence not imply pointwise convergence for continuous functions?

This question shows that $L^2$ convergence does not show pointwise convergence, even when the functions involved are continuous. This strongly contradicts my intuition, because I thought that weak-$L^...
-1
votes
2answers
54 views

Why does $i = 0$ or $\tau = 0$ following this logic? [duplicate]

Following this logic, $i$ appears to equal $0$: $$e^{i\theta} = \cos(\theta) + i\sin(\theta)$$ $$e^{i\tau} = \cos(\tau) + i\sin(\tau)$$ $$e^{i\tau} = 1 + i(0)$$ $$e^{i\tau} = 1$$ $$e^{i\tau} = e^0$$ ...
0
votes
1answer
26 views

Incorrect derivation of geometric series

So this is something really really simple but for some reason I honestly cannot figure out why this is wrong. I was deriving the equation of the summation of a geometric series to the nth term ...
1
vote
1answer
132 views

Fermat's last theorem short proof attempt

Fermat's last theorem states: (1) $x^n + y^n = z^n$ has no solutions for x, y, z and n positive coprime integers and n > 2. An open question is whether there exists a simple proof hinted at by ...
4
votes
1answer
56 views

False proof that $\langle\chi,1_G\rangle$ need not be an integer.

I'd like to know where the following calculation has gone wrong. I'm sure it is a silly error. Let $G$ be a finite group acting on the right cosets $G/H$ of $H\le G$. Let $\chi$ be the character of ...
3
votes
1answer
43 views

Cohomology of free groups topologically

I'm trying to see an example of the topological interpretation of group cohomology, with the free group $F(S)$ on a set $S$ of generators, with coefficients in $\mathbb{Z}$ (on which we act trivially),...
7
votes
1answer
367 views

Careless Mathematical Induction Fallacy

This fallacy is given in Bartle's Introduction to Real Analysis (page 15) and I am trying to figure out where the problem is in the "fake proof". Here we are using $\mathbb{N} = \{1,2,3 \dots \} $ ...
1
vote
2answers
39 views

Hint for proving $\text{Var}\left(\frac{1}{n}\sum_{i=1}^{n}X_i\right)=ρσ^2+\frac{1-ρ}{n}σ^2$

So this is part of a homework at university, which means, I obviously don't want a complete solution as answer, but only a hint as to what I do wrong. We can assume $X_i$ to be one of $n$ identically ...
1
vote
1answer
52 views

What's wrong with my proof of quotient rings?

I seem to have arrived at a contradiction by applying what I know about quotient rings. I can't figure out where the mistake is. Let $f(x)\in \mathbb{Z}[x]$, $\deg f(x)\ge 1$, and $p$ a prime number. ...
1
vote
2answers
60 views

Error in the derivative proof

Assume $h$ is a function on an open interval $K$, and differentiable on $K$. Therefore $h'$ is cont on $K$. The faulty proof goes as follows: Let $a\in K$ By the definition of the derivative: $$h'(...
-1
votes
3answers
86 views

Does $\sqrt{-1\cdot-1}=1$ or $-1$? [duplicate]

Let's define $x=\sqrt{ab}$, where $a=-1$ and $b=-1$. Does $x=1$, as $-1\cdot-1=1\implies\sqrt{-1\cdot-1}=\sqrt{1}=1$? Or maybe $x=-1$, as $\sqrt{a^2}=a\implies\sqrt{-1\cdot-1}=\sqrt{(-1)^2}=-1$? I ...
0
votes
2answers
29 views

Difficutly recognising flaw in putative theorem

I'm trying to answer a textbook question from page 188 of How To Prove It, second edition by Daniel J. Velleman, but I can't figure it out. Consider the following putative theorem. Theorem? Suppose $...
3
votes
1answer
47 views

Help in finding the fallacy in the following proof.

I'm currently studying Michael Spivak's Calculus and stumbled across a problem. I seemed to have proven a ridiculous statement: Let $f$ be continuous over $\mathbb{R}$. Then given sufficiently small, ...
0
votes
1answer
169 views

Why still people are searching for elementary proof of Fermat's Last Theorem?

I was searching in SE and Google for elementary proofs of Fermat's Last Theorem (FLT) and I found a lot of false claims about an elementary proof is found for FLT. I'm wondering: why still there are ...
-1
votes
6answers
159 views

Proof that $0=1$? [duplicate]

I recently saw the following "proof" online, and couldn't pinpoint where the mistake was made: From a well known property, $$1+2+3+\cdots = -\frac{1}{12}.$$ Multiplying both sides by $-1,$ we get $$-...
1
vote
0answers
36 views

Confusion about infinite product of fields

Consider $R=\Pi_{i\in \mathbb{N}} k$, where $k$ is a field. We know that its Spectrum $Spec(R)$ is quasi-compact. Moreover, by many answers already on this site, we know that the Krull dimension of $R$...
2
votes
3answers
98 views

Proof that the square root of a negative number is real.

So I stumbled upon this weird result when experimenting with fractional exponents. Suppose you have some negative, real number, for example -8. We know $\sqrt{-8}$ is not a real number.But $$\sqrt{-8} ...
4
votes
1answer
76 views

What's wrong with my deformation retract?

I drew a deformation retract of the twice punctured genus 2-torus. It would seem that this should deformation retract to a bouquet of 3 circles. One the other hand, computing the Euler-...
0
votes
6answers
83 views

Where is the logical flaw in solving this equation?

I ran across this equation... $\sqrt {2x+6}+4=x+3$ Without thinking, I solved for x in the following way: $\sqrt {2x+6}+4=x+3$ Subtract 4 from both sides. $\sqrt {2x+6}=x-1$ Square each side. ...
0
votes
2answers
80 views

Fake proof that 1 = -1 (Can't understand the mistake) [duplicate]

$1 = \sqrt{1} = \sqrt{(-1)^2} = \sqrt{-1}\sqrt{-1} = i\cdot i = -1$. I know the mistake is here $\sqrt{(-1)^2} = \sqrt{-1}\sqrt{-1}$ because everything else seems right to me, but I don't understand ...
3
votes
2answers
50 views

Proof by deduction - implications

Currently trying to explain some maths to a friend. He has taken a statement $x^2 + 4 > 2x$ and tried to prove this is true for all $x$. His proof is $x^2+4>2x \Rightarrow x^2-2x + 4 > 0 \...
3
votes
1answer
56 views

Locally free $\mathcal{O}_{X}$ modules are not projective

Let $(X , \mathcal{O}_{X})$ be a locally ringed space. I know that in general it is not true that locally free $\mathcal{O}_{X}$ modules are projective in the category of $\mathcal{O}_{X}$ modules (...
0
votes
3answers
86 views

What's the error in my proof of the statement: The product of two irrational numbers is irrational?

Statement: The product of any two irrational numbers is irrational. Formally it can be written as: $$\Big(\forall x \forall y\Big)\,\Big(\big(x \notin \mathbb{Q} \wedge y \notin \mathbb{Q}\big) \to \...
0
votes
1answer
60 views

Why do we need continuity of partial derivatives to prove differentiability?

I was reading the following theorem on Apostol's Mathematical Analysis (page 357): Assume that one of the partial derivatives $D_1f\dots D_nf$ exists at $\mathbf c$ and that the remaining $n−1$ ...