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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4
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3answers
106 views

Proof that any number is equal to $1$

Before I embark on this bizzare proof, I will quickly evaluate the following infinite square root; this will aid us in future calculations and working: Consider $$x=\sqrt{2+\sqrt{{2}+\sqrt{{2}+\sqrt{{...
0
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1answer
48 views

A paradox in surds and $i$. [duplicate]

I have been solving some problems and as part of some manipulation in an expression I had to square the following: $$\sqrt{-1}$$which is generally and famously known as $i$. Before I explain my ...
5
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0answers
158 views

Nelson's proof of the inconsistency of arithmetic

What are the prerequisites to understand Nelson's attempted proof of the inconsistency of arithmetic and the subsequent "famous" discussion on John Baez's blog? I have a reasonable ...
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2answers
96 views

About Cohen's proof for Goldbach's conjecture

As a context, my advisor recently lent me the book Uncle Petros & Goldbach's Conjecture to read, a story about a man obsessed trying to prove/disprove the Goldbach conjecture. I was searching ...
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1answer
58 views

Is Hausdorff maximal principle and Kuratowski lemma same things? [closed]

In the John's Kelley "General Topology" (pages 32-34) there is definition of both of them and its look like its just different ways of notation of one proposition. But also there is ...
2
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1answer
27 views

I somehow deduced that $\tan x=\iota$ for any real value of $x$ by equating the value of $\tan(\frac{\pi}{2}+x)$ obtained using two identities.

Let's assume that we're familiar with the identity : $\tan \Bigg (\dfrac{\pi}{2} + x \Bigg ) = -\cot x$ which we have derived using the unit circle. I was trying to equate the values of $\tan \Bigg ( ...
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1answer
87 views

What is wrong with this proof that P≠NP?

Proof by contradiction: Assume that $P=NP$. There must exist a function that maps from NP to P. So there must exist a function $T$ that, given the source code of a polynomial-time verifier function $V$...
-4
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1answer
33 views

Factoring cos, and other trig functions

EDIT: I misattributed the solution to factoring. The teacher in fact used a the trig identity: $\cos(A+B) = \cos(A)\cos(B) - \sin(A)\sin(B)$ I apologise for the time wasted on my expediant attempt at ...
0
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1answer
54 views

What is wrong with this proof that every ideal whose radical is prime is a primary ideal?

In Dummit & Foote, the definition of primary ideal says: A proper ideal of a commutative ring is called primary if whenever $ab \in Q$ and $a \notin Q$, then $b \in {\rm rad}(Q)$. Suppose $I$ is ...
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2answers
128 views

Find the flaw in this proof that $1$ is the greatest natural number [closed]

I think the flaw is in assuming that $N^{2} \in \mathbb{N}$, but I don't know.
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2answers
54 views

How is it possible that if $A \implies B $ is true then $ \lnot ( \lnot B \implies A )$ can be false?

While I was playing around with the material implication I made a proof by contradiction which I think it's wrong, but I don't find any mistake : Say that $A \implies B $ is true , then suppose the ...
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4answers
77 views

Calculus proof: $0=1$ What is my mistake?

The quotient rule states that: $$\frac{d}{dx}\frac{f(x)}{g(x)}=\frac{f'(x)g(x)-f(x)g'(x)}{{g(x)}^2}=\frac{f'(x)}{g(x)}-\frac{f(x)g'(x)}{{g(x)}^2}$$ Integrating tells us that $$\frac{f(x)}{g(x)}=\int\...
2
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1answer
65 views

$\int_0^x g(x-a)f(a)da=\int_0^x f(a)da$ - Please help to explain the error

Let $I$ be an integral function s.t. $$I(f(x))= \int_0^x f(a)\,da.$$ Let $g$ be a function s.t. $g(0)=1$ and $g_x(a):=g(x-a)$. I have \begin{align} \int_0^x g_x(a)f(a)\,da& =I(g_x(x)f(x)) \\[...
2
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1answer
56 views

Where is the error? Application of FTC

Please, I need a help to see the error on this argument:$$\int_0^tf(a)g(t-a)da=\int_0^tf(t-a)g(a)da\implies$$ $$\dfrac{d}{dt}\int_0^tf(a)g(t-a)da=\dfrac{d}{dt}\int_0^tf(t-a)g(a)da\implies$$ $$f(t)g(...
3
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1answer
143 views

Whats the error in my continuum hypothesis “proof”

First of all, I just want to say that I know my "proof" is incorrect due to the continuum hypothesis being unprovable using the standard ZFC axioms. The reason I'm posting this is because I'm self ...
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0answers
25 views

False proof about $\int_{0}^{\infty}\frac{\sin^2(x)}{x^2}dx=\frac{\pi}{2}$

We have : $$\int_{0}^{\infty}\frac{1}{\cosh(x)}dx=\frac{\pi}{2}$$ There exists a simple antiderivative wich is : $$2\tan^{-1}\Big(\tanh\Big(\frac{x}{2}\Big)\Big)$$ I recall that : ...
3
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1answer
38 views

Why shift operator is not homotopic to 1 ($K_1$-approach)?

Let us recall that via fourier transform it holds true that $C^*(S)\cong C(\mathbb{T})$, with map given by $S\mapsto e^{2\pi i x}$ (considering $\mathbb{T}=\mathbb{R}/\mathbb{Z}$). It is also true ...
1
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1answer
31 views

Fake Proof Regarding Factorization of Binomial Coefficients

What is wrong with the following "proof" that ${{n}\choose{r}}={{n}\choose{k}}\cdot{{n-k}\choose{r-k}}$ where $k<r$? Each possible combination can be divided into two disjoint sets, the first ...
7
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2answers
215 views

Invertibility of elements in $A[x]$ with coefficients in the Jacobson radical

While solving an exercise about invertibility of elements in a polynomial ring, I came up with the following "proof" that a polynomial is invertible if its zeroth coefficient is invertible and all ...
3
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1answer
46 views

Compactness theorem in modal logic

There is a straightforward proof of the compactness theorem for propositional logic (see here) involving the following steps: Start with a finitely satisfiable set Extend the set into a set that ...
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0answers
37 views

Grandi's series manipulation allowing me to prove 1 = 0?

Let me start by saying I have a very basic background in math, so if I'm using the wrong terms, please feel free to point it out. I know straight away that this is wrong, I'm just curious to know ...
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0answers
46 views

Find the mistake: $A=\emptyset\land B=\emptyset\leftrightarrow A\times B=\emptyset$

Find the mistake that for all $A$ and $B$, $$A=\emptyset\land B=\emptyset\leftrightarrow A\times B=\emptyset.$$ The "proof": $$ \begin{align*} A=\emptyset\land B=\emptyset&\leftrightarrow\neg(\...
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1answer
43 views

different answers for same division

I recently came across an easy question which stumped me . It is as follows: simplify the following expression a/b/c/d/e/f . My approach: for a/b/c/d it's easy to see as dividing a fraction (a/b) ...
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2answers
43 views

Are two functions linearly dependent iff their Wronskian is $0$?

The fact that linear dependence guarantees that $W=0$ is easy to prove. What about the converse? According to Wikipedia its not necessarily true. But, what is the problem is this proof? $$W=y_1y_2'-...
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2answers
53 views

Showing that we can always decompose $Y = \mathbb{E}\left[Y|X\right] + \epsilon$.

The question is that Let $Y$ be a random variable with finite mean and $X$ a random vector. $Y$ can always be decomposed as follows: $Y = \mathbb{E}\left[Y|X\right] +\epsilon$, where $\epsilon$ is ...
4
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1answer
73 views

What's wrong with this proof? if $\sum a_n$ converges and $\sum b_n$ converges absolutely, then $\sum a_nb_n$ converges

Proof: Since $\sum a_n$ converges, then $a_n \to 0 $ so there exists $M > 0$ such that $a_n < M$ for all $n$. Also, there exists $N$ such that for all $m,n \geq N$ we get $|\sum_{k=n}^m |b_k||...
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1answer
32 views

Find the mistake in the following proof by induction (exercise in AOC, Vol.1, Knuth)

In the Art Of Programming by Knuth there is the following exercise: There must be something wrong with the following proof. What is it? "Theorem. Let $a$ be any positive number. For all positive ...
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1answer
35 views

Find the mistake on the “proof” that $\exists x(\neg p(x)\land\neg q(x))\implies\exists x\,\neg p(x)\land\exists x\,\neg q(x)$

Find the mistake on the "proof" that $$\exists x(\neg p(x)\land\neg q(x))\implies\exists x\,\neg p(x)\land\exists x\,\neg q(x).$$ "Proof": $$ \begin{array}{lll} 1)&\exists x(\neg p(x)\land\neg q(...
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1answer
65 views

Doubt in IMO 1982 problem 1

Question - The function $f(n)$ is defined for all positive integers $n$ and takes on non-negative integer values. Also, for all $m, n$ $$ \begin{array}{c} f(m+n)-f(m)-f(n)=0 \text { or } 1 \\ f(2)=0,...
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2answers
67 views

Likely fake proof of the irrationality of a rational multiple of pi/pi

I would like to know what has gone wrong in this 'proof'. Suppose that $$k\frac 1\pi\pi=\frac ab\operatorname{,where}a,b\in \mathbb Z\operatorname {and}k\in \mathbb Q.$$ Then, we multiply both sides ...
0
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1answer
44 views

The power of induction.

I was thinking about the triangle inequality for complex values, and was wondering if this would be a sufficient proof. In attempts to show the triangle inequality for complex numbers, that is $$|z+w|\...
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2answers
46 views

Prove that $\; U\setminus A = U \iff A=\emptyset\; $ where $U$ is the universe (False proof of the opposite).

I think I might have made a false proof for the following problem but I can´t seem to find where the flaw is. Prove that $\displaystyle \; U\setminus A = U \iff A=\emptyset\; $ where $U$ is the ...
4
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1answer
37 views

What is wrong with this fake proof that any subset of a $T_1$ space is closed?

Let $X$ be a $T_1$ space and let $A \subseteq X$ be any subset. Fake Proof: Since $X$ is $T_1$ we know that any singleton in $X$ is closed. So choose $a \in A$, then $\{a\}$ is closed and $X \...
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1answer
15 views

How estimate series by its own term in the Banch space?

Let $(X, \|\cdot\|)$ be a Bench space. Assume that $x_n \in X$ for all $n=0,1,....$ We may notice that $\|f\|\geq \|g\|-\|f-g\| $ for $f, g\in X.$ Put $y= \sum_{n=0}^{\infty} x_n.$ Notice that $\|y\...
0
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1answer
53 views

Why is it so difficult to prove the irrationality of the zeta function for $N$?

I know that no one has yet been able to prove that the zeta function is irrational at every point $N$. but my question is why is it so difficult to prove the irrationality of the zeta function for $N$?...
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1answer
43 views

Understanding where my naive attempt to prove Countable Choice out of Finite Choice fails

I am aware of this and this topic, but I would like to receive a clarification concerning the foregoing naive attempt to prove Countable Choice, to see if I understood properly the question. Assume ...
0
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1answer
46 views

Why is P(A⋃B) ⊄ P(A)⋃P(B)?

I had a question in my 11th grade Mathematics book : Prove that P(A⋃B) ⊄ P(A)⋃P(B) I did a proof for why P(A⋃B) ⊂ P(A)⋃P(B) Which step in the following proof is wrong? ...
4
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4answers
102 views

Is $\sin(\alpha)=\frac{\tan(\alpha)}{\sqrt{1+\tan^2(\alpha)}}$ a true statment?

I'm being asked to prove the following equality $$\sin(\alpha)=\frac{\tan(\alpha)}{\sqrt{1+\tan^2(\alpha)}}$$ and I support the idea that they are not equal (the rest of my class seems to desagree ...
4
votes
1answer
100 views

Can't find why the proof is false (substitution rule).

Statement: Suppose $f$ and $g$ are functions with domain $\mathbb{R}$ and we want to show that if $$\lim_{x \rightarrow a} f(x) = b\ \ \text{and}\ \lim_{x \rightarrow b} g(x) = c,\ \text{then}\ \...
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0answers
20 views

Paradox curve lengths limit

A possible similarity link Consider a sequence of curves as follow: $C_1$ is a semicircle drawn on the segment $[0,1]$, as in figure bellow; $C_2$ is the union of semicirocles of diameters $[0,1/2], ...
0
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2answers
60 views

Which step is wrong? [duplicate]

(1)$1=\sqrt{1}$ (2)$=\sqrt{-1×-1}$ (3)$=\sqrt{-1}×\sqrt{-1}$ (4)$=i×i=-1$ $1=-1$ Which step is wrong? I guess step 3 but I dont know why
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0answers
45 views

is my proof by contradiction correct?

Suppose that n is a positive even integer with $n/2$ being odd. Prove that there do not exist positive integers $x$ and $y$ with $x^2−y^2=n$. Assume that there exist positive integers positive ...
0
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2answers
78 views

If $g$ and $g\circ f$ are continuous function, then $f$ is a continuous function?

Let be $g:Y\rightarrow Z$ a continuous function beetween two topological spaces and we suppose that the function $f:X\rightarrow Y$ beetween the topological spaces $X$ and $Y$ is such that its ...
0
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2answers
43 views

Roots, polynomials and equality breaking

Suppose I have $$ z=(a+ib)\implies z^2=(a+ib)^2\implies z=\pm(a+ib) $$ But if I do this instead: $$ \begin{align} z&=a+bi\\ \implies z^2&=a^2-b^2+2iab\\ &=a(2a+2ib-a)-b^2\\ &=a(2z-a)...
0
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2answers
52 views

How to prove that $\mathscr{cl}(X\setminus A)\cap X\setminus\mathscr{cl}(B)\subseteq\mathscr{cl}(X\setminus A\cap X\setminus B)$?

To prove the assertion I attempt two different way: but unfortunately both seem inconclusive. Below you can read the two different "demonstration". $B\subseteq\mathscr{cl}(B)\Rightarrow X\setminus\...
2
votes
1answer
50 views

The value of $(-0.1)^{-0.1}$

I saw a video about how the answer for this is complex because- $(-0.1)^{-0.1}$ $\frac{1}{(-0.1)^{0.1}}$ $\frac{1}{(-0.1)^\frac{1}{10}}$ $\frac{1}{\sqrt[10]{-0.1}}$ $\sqrt[10]{-0.1} \;\epsilon \;\...
0
votes
0answers
40 views

Resolving a gap in the Kernel proof of the Dirichlet Integral

One well known proof of the Dirichlet Integral is to use the kernel $\frac{\sin{(2n+1)x}}{\sin{x}}$ and applying the Riemann-Lebesgue Lemma to an auxiliary function to obtain an equivalence between ...
0
votes
0answers
63 views

Is this a proof?

Suppose someone does research on a phenomenon and builds the model $y=a+bx$. The scientist then states the "Theorem" that, given the particular assumptions leading to the model, $x$ affects $y$ ...
1
vote
1answer
67 views

Bernoulli number generating function proof problem

A sequence $(b_n)_n$ is given as $b_0 = 1$ and for every $n \in \mathbb N$ $$\sum_{k=0}^n\binom{n+1}{k}b_k=0\tag{1}$$ The task is to find its exponential generating function $f(x)$ From $(1)$ we get $...
2
votes
2answers
60 views

Finding the mistake in induction proof

This argument seems obviously wrong, but I can't find the mistake. Any ideas? There are two people, each given a positive integer, such that the two integers differ by $1$. The rule of the game ...

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