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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.

2
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1answer
44 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
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2answers
30 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
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2answers
33 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
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1answer
66 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^...
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2answers
45 views

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

Following this logic, $i$ appears to equal $0$: $$e^{i\theta} = \cos(\theta) + i\sin(\theta)$$ $$e^{i\tau} = \cos(\tau) + i\sin(\tau)$$ $$e^{i\tau} = 1 + i(0)$$ $$e^{i\tau} = 1$$ $$e^{i\tau} = e^0$$ ...
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1answer
23 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
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1answer
114 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
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1answer
54 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
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1answer
40 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
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1answer
352 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 \} $ ...
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2answers
36 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
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1answer
49 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
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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'(...
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3answers
82 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
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2answers
26 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
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1answer
46 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
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1answer
150 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
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6answers
147 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
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0answers
28 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
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3answers
93 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} ...
5
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1answer
64 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
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6answers
48 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. ...
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2answers
75 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
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2answers
47 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
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1answer
47 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
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3answers
81 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
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1answer
44 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 $n−1$ ...
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0answers
65 views

Finding an inconsistency in a seemingly flawless elementary proof of FLT

In an elementary "proof" (most likely, a wrong one) of Fermat's Last Theorem, I spotted an inconsistency I want you to confirm. I know, I am probably wasting my time doing this, but still, spotting a ...
4
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3answers
49 views

Why the opposite direction of proving $\mathcal P(A) \cup \mathcal P(B) \subseteq \mathcal P(A\cup B)$ is wrong

The following proof is given to show that $\mathcal P(A) \cup \mathcal P(B) \subseteq \mathcal P(A\cup B)$ ($\mathcal P$ is the power set): Let $X \in (\mathcal{P}(A) \cup \mathcal{P}(B))$, by ...
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7answers
372 views

Elementary question about a fake-proof and greatest common divisors

I have a question to an excercise - for which I have a wrong solution - and I wanted to ask you to help me understand my thinking error. The excercise was as follows: Let $a, b, n \in \mathbb{N}$. ...
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3answers
2k views

Where is the mistake in this “proof” of the inconsistency of ZFC?

This is a "proof" that ZFC is inconsistent, but I haven't found the mistake yet. Let $\{\varphi_n \colon n <\omega\}$ be an enumeration of all formulas in $L_{\in}$ with exactly one free ...
0
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5answers
102 views

Starting with a false statement, how can one prove anything is true? [duplicate]

So I've been learning a bit of logic for class and heard that if you begin with a false statement, you can then prove anything to be true, however I don't entirely understand what this means or how to ...
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2answers
76 views

What went wrong in this fake proof?

So, I came up with this quite ridiculous fake proof that no nonnegative integers $(x,y)$ satisfy $x^2+y^2=5$. Clearly, $(x,y)=(1,2)$ satisfies the equation, so the proof is wrong; but somehow, I can't ...
0
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1answer
24 views

check if inequality is real

check if inequality is real: 30n < 2^n + 105 my solution proposal: n=1 30<107 true n=k 30k < 2^k +105 30(k+1) < 2^{k+1} + 105 Proof 30(k+1) < 2^{k+1} + 105 30k+30< 2^k +105+...
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4answers
937 views

What is wrong with this fake proof that subgroup of a cyclic group is cyclic?

Let $G$ be a cyclic group generated by $a$ and $H$ its subgroup. This is a proof by contradiction. Assume there is no $r$ in $H$ such that $\langle r \rangle = H$. If some $s$ in $G$ is not in $H$ ...
6
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1answer
118 views

Why isn't Legendre's Conjecture resolved by the work done by Nair and Hanson in relation to Least Common Multiples?

I recently discovered the work done by Hanson and Nair. As I worked through Hanson's proof, an argument occurred to me regarding Legendre's Conjecture. Clearly, my argument is wrong. It is too ...
0
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1answer
51 views

What's wrong with this induction? (Runtime analysis of standard polynomial long division)

In my studies for my Bachelor's thesis, I've gone through a runtime analysis of plain vanilla polynomial long division, i.e. I wanted to prove the statement: Let $f,\,g \in F[X] \land g\neq 0$ where $...
3
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2answers
73 views

Why does my proof fail to show the logical equivalence of (∀x)(Fx v Gx) ⊢ (∀x)Fx v (∀x)Gx?

Apparently (∀x)(Fx v Gx) is not equivalent to (∀x)Fx v (∀x)Gx, however I seem to be able to prove it syntactically: (∀x)(Fx v Gx) ⊬ (∀x)Fx v (∀x)Gx (1) (∀x)(Fx V Gx)-----premise (2) Fa v Ga----------...
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0answers
53 views

What's the fallacy in $i^{-1} = i$ [duplicate]

We need to express $i^{-1}$ as $a+bi$ where $a,b \in \mathbb{R}$. There are a lot of ways to simplify it. One can easily see that $i^{-1} = i^3= -i$, for example. However, this is not the approach ...
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1answer
59 views

Where does this derivation of the Fourier Series for csc(x) go wrong?

In this post, the following derivation for the Fourier series of csc(x) is given: \begin{align} \csc x &= \dfrac{1}{\sin x}\\ &= \dfrac{2i}{e^{ix}-e^{-ix}}\\ &= \dfrac{2ie^{-ix}}{1-e^{-...
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0answers
32 views

Wrong intuition with partial derivatives.

So I noticed I have some wrong intuition about partial derivatives and I would appreciate someone correcting me why I am wrong. Say we have a function $f(x,y)=2\sqrt{xy}$. Clearly $f_x(x,y)$ ...
42
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1answer
9k views

What is the Todd's function in Atiyah's paper?

In terms of purported proof of Atiyah's Riemann Hypothesis, my question is what is the Todd function that seems to be very important in the proof of Riemann's Hypothesis?
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3answers
81 views

Why are cases required in proofs? i.e. in the inequality $|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|}$?

So there is a proof of the inequality needed: $|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|} $ , where $x$,$y$ $\geq$ $0$ After squaring both sides: $(|\sqrt{x}-\sqrt{y}|)^2 \leq (\sqrt{|x-y|})^2$ $x + y ...
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2answers
41 views

Let $C$ be compact sets. $\bigcap_{i \in K} (C_i)=\emptyset \Rightarrow \exists F \subseteq K \; \text {finite} | \bigcap_{i \in F} (C_i) = \emptyset$

I made a mistake while proving the theorem in the title, however, I don't see where. Here's the theorem: Let $C$ be closed and limited subsets of $\Bbb{R}$. Then $\bigcap_{i \in K} (C_i) = \...
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3answers
225 views

Proof that 2=1 using differentiation [duplicate]

We can express a square number as the repeated addition of that number in this manner: $1^2 = 1$ $2^2 = 2 + 2$ $3^2 = 3 + 3 + 3$ Generalising this, we get: $x^2 = x + x + x...$ $x$ $times$ If we ...
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1answer
25 views

Understanding a mistake regarding removable and essential singularity.

It is a well known fact that $0$ is an essential singularity of the function $e^{1/z}$ therefore it is not essential nor a pole. On the other hand we know the Rieamann's continuation theorem which ...
0
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0answers
58 views

Why is this counterexample to an established result wrong?

It is a well known fact that, for a real function $f$, if $f'(x)$ has a limit as $x\to x_0$, then $f$ is differentiable at $x_0$. In fact, there has been some questions on this site about proving this ...
0
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0answers
133 views

Modern Mathematical Crankery: A List.

I'm sorry if this question is off topic. I'd also like to apologise if this is deemed too broad or otherwise a poor question. The Question: I'm interested in examples of modern mathematical ...
0
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1answer
36 views

Proof of the formula for area of a polar function.

I was wondering what i am doing wrong in my proof of the formula $$A=\int r^2/2$$ First of all we know, given parametric equations $x=f(t)$ and $y=h(t)$ the area represented by that parametric curve ...
18
votes
3answers
2k views

Where is wrong with this fake proof that Gaussian integer is a field?

The Gaussian integer $\mathbb{Z}[i]$ is an Euclidean domain that is not a field, since there is no inverse of $2$. So, where is wrong with the following proof? Fake proof First, note that $\...