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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On correctness of induction proof

I want to prove a certain property $\mathsf{P}$ on every multiaffine polynomial in $\Bbb R[x_1,x_2,\dots,x_{n-1},x_n]$. Supposing I show property $\mathsf{P}$ to be valid at $n\geq9$ variable ...
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 ...
-1
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1answer
170 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
37 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
139 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 ...
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4answers
188 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 ...
0
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3answers
64 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 $$-3x^2+7 = -3y^2+7$$ $$-3x^2 = -3y^2$$ $$x^2 = y^2$$ $$x = y$$ ...
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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*} ...
-3
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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
90 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
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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
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4answers
100 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
70 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
24 views

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
18 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?
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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
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2answers
128 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 ...
-3
votes
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 ...
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2answers
460 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 ...
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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
131 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
705 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
85 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
118 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
53 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
187 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
247 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
82 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
126 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 ...
2
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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??
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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 ...
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2answers
521 views

What is the flaw in this proof that all triangles are isosceles?

What is the flaw in this "proof" that all triangles are isosceles? From the linked page: One well-known illustration of the logical fallacies to which Euclid's methods are vulnerable (or at least ...
5
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3answers
222 views

Proof of $x*0=0$ is invalid?

My professor told me today that while my logic was good and all my steps after the assumption were correct, my argument was invalid because I was assuming what I want to prove. I don't see how. ...