Seemingly flawless arguments are often presented to prove obvious fallacies (such as 0=1). This tag is the appropriate place to ask "Where is the proof wrong?" when encountering such proof.

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45 views

Cantor set countable? [duplicate]

I know the Cantor set is uncountable, but I just came with an argument that shows it is countable. Obviously my argument is wrong, but I just don't know where is the mistake. Here it is. Let $C$ be ...
1
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0answers
28 views

How can I solve this para Paradox? [duplicate]

How can I solve this para Paradox? $ -1={(-1)}^{1/2} {(-1)}^{1/2}={[(-1)(-1)]}^{1/2}=1$
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2answers
81 views

What is wrong with my proof: $-1 = 1$? [duplicate]

I have some theories about why this could by wrong but I still haven't something that convinces me. What is wrong with this proof: $ -1 = i^2 = i.i = \sqrt{-1}.\sqrt{-1} = \sqrt{(-1).(-1)}= \sqrt1 = ...
16
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4answers
2k views

I can't remember a fallacious proof involving integrals and trigonometric identities.

My calc professor once taught us a fallacious proof. I'm hoping someone here can help me remember it. Here's what I know about it: The end result was some variation of 0=1 or 1=2. It involved ...
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1answer
178 views

Proof for Goldbach's Conjecture [closed]

There is a proof given here. I couldn't find any flaw in it, what's wrong with it?
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0answers
27 views

Mistake in proof that a polynomial $f$ irreducible in $F$ is irreducible in $E$ if $\gcd(\deg f, [E:F])=1$

This is a problem in James Milne's text on Galois Theory: Let $f(x)$ be an irreducible polynomial over $F$ of degree $n$, and let $E$ be a field extension of $F$ with $[E:F] = m$. If $\gcd(m,n)= ...
0
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1answer
9 views

Convergence of matrix exponentiation

Let $A$ be a $m\times m$ complex matrix. In my text, it proves the convergence of $\sum \frac{A^n}{n!}$ by using Jordan canonical form which is quite tricky. However, isn't it much easier to prove ...
2
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1answer
38 views

False proof of 0=1 using Laurent series

I found the following proof that 0 = 1: \begin{align*} \sum_{n=-\infty}^{\infty} 0\cdot z^n = 0 = \frac{1}{z-1} + \frac{1}{1-z} = \frac{1}{z}\frac{1}{1-\frac{1}{z}} + \frac{1}{1-z} \\ = \frac{1}{z} ...
1
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1answer
140 views

Fake $0=1$ integral examples.

The classic "proof" that says 0=1 with integration by parts is this: $$\int\frac{1}{x}\,dx=x\frac{1}{x}-\int -\frac 1{x^2}x\,dx=1+\int \frac1x\,dx.$$ However the wikipedia article gives another one of ...
7
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4answers
189 views

Why doesn't every integral from 0 to $2\pi$ equal zero?

Quick question, might end up having a simple answer, but I have here a "proof" that any integral from 0 to $2\pi$ is zero, as follows: $$\int^{2\pi}_0f(x)dx$$ Now using u-substitution, let $u = \sin ...
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3answers
65 views

Whats wrong with this proof? (trying to prove a function is surjective)

Let $f : A → B$ and $g : A' → B'$ be functions that are onto, and $h : A × A' → B × B'$ be the function $h(x, y) = (f(x), g(y))$. Part1. Dr. Bob tries to prove that $h$ is onto with the following ...
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2answers
51 views

Fake proof of one-to-one function

Prove or disprove that $f: \mathbb{R} \rightarrow \mathbb{R}$ is one-to-one if, $$f(x) = -3x^2+7$$ Assume $f(x) = f(y)$, then $$-3x^2+7 = -3y^2+7$$ $$-3x^2 = -3y^2$$ $$x^2 = y^2$$ $$x = y$$ ...
7
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2answers
135 views

Integrate $1/x$ by parts.

$$\int \frac{\mathrm{d}x}{x}$$ If I integrate this by parts ($\displaystyle u=\frac{1}{x}, \mathrm{d}u = -\frac{\mathrm{d}x}{x^2}, \mathrm{d}v= \mathrm{d}x, v = x$), then why does this happen? $$\int ...
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6answers
144 views

Does $\frac{0}{0}$ really equal $1$? [duplicate]

If we agree that $\textbf{(a) }\dfrac{x}{x}=1$, $\textbf{(b) }\dfrac{0}{x}=0$, and that $\textbf{(c) }\dfrac{x}{0}=\infty^{\large\dagger}$, and let us suppose $z=0$: $$\begin{align*} ...
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1answer
57 views

fake proof of $\forall a. \forall b. a = b \to 1 = 0$

I saw a less formal version of this fake proof that claimed to prove $2=1$ but because it assumed $a=b$ from the start I knew why it was wrong. It does seem however that the proof can be used to prove ...
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2answers
92 views

What did I do here? This can't be right… ($i = -1$)?

I was messing around in Geometry class today and found a very odd 'proof'. It relies on only two facts, $1^2=1$ and $i=\sqrt{-1}$ From here I did this: $$i = \sqrt{-1}$$ $$i^2 = -1$$ $$-i^2 = 1$$ ...
5
votes
1answer
564 views

Can someone point out the flaw in my proof of AC?

I have a fake proof of the axiom of countable choice. Obviously it is not correct, but I cannot see my flaw. Forgive me, I am only learning set theory. Let $\{A_n : n \in \mathbb{N}\}$ be a countable ...
3
votes
4answers
101 views

how to point out errors in proof by induction

I have searched for an answer to my question but no one seems to be talking about this particular matter.. I will use the all horses are the same color paradox as an example. Everyone points out ...
0
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1answer
71 views

What is the error in this fake-proof of the complex number i? [duplicate]

The error is from the 3rd step to the 4th step. But why is this an error? Can't $i$ be interchangeable with $\sqrt{-1}$? $-1 = i\cdot i = \sqrt{-1}\cdot \sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{1} = 1$
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0answers
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Question from Stewart's Calculus regarding proof of independence of path and conservative vector fields.

Please look over this proof. In the proof, it says: "Notice that the first of these integrals does not depend on $x$, so..." How is that so? $C_1$ does depend on $x$. How/Why does it not ...
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1answer
19 views

For $f(\theta)= e^{\theta}$. Is it true that $\hat{f}(n)(1-in)=0$ for all $n\in \mathbb Z.$?

(This is motivated from the following question) Fact: If $f \in C^1(\mathbb{T})$, then the Fourier coefficients $\widehat{f'}(n)$ of the derivative $f′$ can be expressed in terms of the Fourier ...
2
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3answers
167 views

$1^x = 1^y$ and $x,y$ belongs to Real Numbers.

$1^x = 1^y$, and $x,y \in \mathbb{R}$. Following the rule, same base has powers equal every $x$ should be equal to every $y$. $$1^x = 1^y$$ $$x = y$$ What went wrong?
0
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2answers
51 views

what are the problems with the followings “equations”?

what are the problems with the followings "equations"? A) In the complex number field consider the following: $-1=i^2=(i^4)^{\frac{1}{2}}=1^{\frac{1}{2}}=1$. B) In $\Bbb R$, ...
3
votes
2answers
129 views

“Proof” that $0=1$ using an integral. [duplicate]

I saw the following: $$\begin{align} \int \tan x \ \mathrm{d}x &= \int \sin x \sec x \ \mathrm{d}x \\ \int \tan x \ \mathrm{d}x &= -\cos x \sec x - \int - \cos x \sec x \tan x \ \mathrm{d}x ...
1
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0answers
139 views

Derivative of conditional expectation

Let $\left( {{X_t}:t \in \left[ 0 \right.\left. {, + \infty } \right\rangle } \right)$ be a continuous time Markov chain on a probability space $\left( {\Omega ,\mathcal{F},\mathbb{P}} \right)$ with a ...
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1answer
42 views

Question on series $\frac {\Gamma'(z)}{\Gamma(z)}$

Prove that: $$\frac {2\Gamma'(2z)}{\Gamma(2z)}-\frac {\Gamma'(z)}{\Gamma(z)}-\frac {\Gamma \prime(z+\frac{1}{2})}{\Gamma(z+\frac{1}{2})} =2 \log 2$$ But I obtain this equal zero: $$\frac ...
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1answer
106 views

Why is this $0 = 1$ proof wrong? [duplicate]

$0 = 0 + 0 + 0 + ...$ $0 = (1 - 1) + (1 - 1) + (1 - 1) + ...$ $0 = 1 - 1 + 1 - 1 + 1 - 1 + ...$ $0 = 1 + (-1 + 1) + (-1 + 1) + (-1 + ...$ $0 = 1 + 0 + 0 + 0 + ...$ $0 = 1$ I can't ...
5
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2answers
465 views

What is wrong with this induction proof?

What is wrong with this "proof" by strong induction? "Theorem": For every non-negative integer $n, 5n = 0$. Basis Step: $5(0) = 0$ Inductive Step: Suppose that $5j = 0$ for all non-negative integers ...
0
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2answers
84 views

Prove $-(-a) = a$

Let $F$ be a field and $a \in F$. Prove $-(-a) = a$. So we want to show that $(-a) + (-(-a)) = 0$, since inverses are unique (I successfully proved that inverses are unique in an earlier problem ...
2
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4answers
133 views

What is wrong with the following “proof” that $e=1$?

Let's analyze this expression $\lim_{n\rightarrow\infty} (1+\frac{1}{n})^n$ It's the definition of $e$ which, as we know is not equal to $1$. So what is wrong with the following "logic": As ...
2
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1answer
37 views

Determine whether $\phi$ is a homomorphism

Let $\phi: \mathbb{Z}_6 \rightarrow \mathbb{Z}_2$ be given by $\phi(x)=$the remainder of $x$ when divided by $2$, as in the division algorithm. Let $\phi: \mathbb{Z}_9 \rightarrow ...
14
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2answers
711 views

What's wrong with this proof that commutativity is implied by the other field axioms?

I seem to have found a proof that the commutativity of $+$ follows from the other field axioms. It is as follows: Let $(k,+,\cdot)$ be a structure satisfying all field axioms except commutativity of ...
0
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2answers
86 views

Proof by induction failure if assumption is wrong?

I never got a clear answer to this question in college. What happens in an induction proof if the assumption is wrong? For example, suppose we try to prove that $n^5$ > n! for n >= $2$ so we start ...
2
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1answer
59 views

Is this a proper way to prove simple geometrical result?

I found this on Quora : Is there anything wrong in the steps illustrated ?
3
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1answer
119 views

What is the incorrect proof by Euler that $\pi = 0$ (or something like that)?

I seem to remember a proof by Euler, involving infinite series, which was really complex (for a maths hobbyist). I believe it was sent in a letter to someone, and that it ended up with $\pi = 0$ or ...
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2answers
45 views

Is this a valid method of proof?

We are given that $y = a + b$, and we want to prove that $y = a + c$ (using all the usual properties of numbers that we know from grade school). Does it suffice to set $a + b = a + c$, and by ...
0
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1answer
61 views

Spotting mistake: unnecessary given condition

I have solved the following problem without using a given premise. Could someone please spot whether I have done something wrong? Suppose we have a relation $\geq$ that is transitive, but not ...
0
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0answers
55 views

Proof of law of reflection using Fermat's principle : are we really proving the law of reflection?

Before you skip reading this, let me tell you that this isn't a "how to derive the law of reflection using Fermat's principle" question. Also, I asked it on MSE instead of the physics site because ...
2
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1answer
48 views

Isn't that proof going the wrong way?

I'm currently working on the very well written book Understanding Analysis, by Stephen Abbott. But I found a proof that looks wrong, I think that it going the wrong way (showing that A $\implies$ B ...
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3answers
188 views

What is wrong with this proof that 3 is less than 1?

What is wrong with this proof? Theorem. 3 is less than 1. Proof. Every number is either less than 1 or greater than 1 or equals 1. Let $c$ be an arbitrary number. Therefore, it is less than 1 or ...
3
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1answer
249 views

Did I construct an infinite set equal to $\{1\}$?

Okay, I'm trying to understand the argument that NJ Wildberger gives in the following video: https://www.youtube.com/watch?v=5CiiGdaYEPU He tries to explain why he things infinite sets don't make ...
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1answer
83 views

Find the mistake in this proof.

I need help with finding the mistake in this proof. Statement: All natural numbers are divisible by 3. Proof: Suppose, for the sake of contradiction, the statement were false. Let X be the set of ...
3
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1answer
128 views

Partial Proof of Second Hardy-Littlewood Conjecture (modified)?

I have obtained the following partial proof of the Second Hardy-Littlewood Conjecture of which I can't find out the logical flaw in the proof. Problem Prove that for all sufficiently large ...
3
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2answers
295 views

Homomorphism problem gone wrong

Okay, so I'm working on a homework problem in abstract algebra, and I have found the solution already, what I want to know is why my initial line of reasoning didn't work - i..e, what have I done or ...
0
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1answer
62 views

Proving property for all predicates in first order logic

Let's consider language with predicate $P$ and following derivation $${{{[P[a/x]]^1} \over {P[a/x] \rightarrow P[a/x]}}\rightarrow I^1 \over {\forall_x (P(x) \rightarrow P(x))}}\forall I$$ Doesn't ...
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2answers
103 views

Proof that $\sqrt{4}\notin\mathbb{Q}$ of course wrong but where is the flaw?

Assume $$\eqalign{ \sqrt{4}\in\mathbb{Q}&\Longrightarrow(\exists a,b\in\mathbb{Z})\sqrt{4}=\frac{a}{b}\text{ and }\gcd(a,b)=1\\ &\Longrightarrow 4b^2=a^2\Longrightarrow a\text{ is even}\\ ...
0
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1answer
87 views

how $2x=x$ , related to differential calculus [duplicate]

can anybody please tell me what's happening here ? $$1^2=1$$ $$2^2=2+2$$ $$3^2=3+3+3$$ $$x^2 = x+x+\cdots+x \mbox{ ($x$ times)}$$ differentiating both the sides $$2x = 1 + 1 + \cdots+1 \mbox{ ...
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1answer
59 views

Apparently same probability questions with different answers.

I was reading A first course in probability by Sheldon Ross when and then I came up with this question. This is how he introduces the famous problem of points Independent trails, resulting in a ...
0
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2answers
103 views

How can be 1 is equal to 2?

It may be a silly question. But I don't know it. So I'm questioning. Recently I've got a proof that proves 1=2. Is there any fault in the proof? If so then what is the fault??
25
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2answers
5k views

Demonstration that 0 = 1 [duplicate]

I have been proposed this enigma, but can't solve it. So here it is: $$\begin{align} e^{2 \pi i n} &= 1 \quad \forall n \in \mathbb{N} && (\times e) \tag{0} \\ e^{2 \pi i n + 1} &= e ...