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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1answer
41 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
71 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 ...
1
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2answers
30 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
55 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
25 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
41 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 ...
1
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3answers
154 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
votes
1answer
135 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 ...
1
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1answer
59 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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2answers
257 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
44 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
93 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
81 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{ ...
1
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1answer
49 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
93 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??
0
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1answer
35 views

How to prove the deduction theorem in a natural deduction calculus?

In van Dalen's Logic and Structure, after defining the notion of a derivation (p.35-6) and syntactic consequence (p.36) the author immediately exhibits some lemmas (1.4.3) where the deduction theorem, ...
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 ...
5
votes
2answers
210 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
votes
3answers
212 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. ...
1
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1answer
43 views

Why does this proof fail?[convergence of infinite sums]

An equivalent way of saying that a normed vector space is complete is saying that every absolutely convergent series, converges. Hence' in some normed vector-space(incomplete), there must be a ...
3
votes
5answers
147 views

'Proof ' that $\ln(x)$ converges

Where is the flaw in the following 'proof '? $$\lim_{x \to \infty}\left[\frac{\mathrm{d}}{\mathrm{d}x}\left\{\ln(x)\right\}\right]=\lim_{x \to \infty}\left[\frac{1}{x}\right]=0 \implies\lim_{x \to ...
2
votes
1answer
84 views

A possible incorrect application of Law of Large numbers

A friend left this teaser for me. He asked me to first compute: $$ \lim_{n \to \infty} \frac{\binom{2n}{n}}{2^{2n}}$$ Using Stirling's approximation (and another method), I got the answer as $0$. ...
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votes
2answers
70 views

What's wrong with this induction based proof?

Claim: $\forall x \in \mathbb{R^+} ,$ $ x^n=1 $ $where$ $ n\in \mathbb{N}$ Proof by induction on n: Basis step: $\forall x \in \mathbb{R^+} ,$ $ x^0=1 $ Induction Step: Let this holds for all ...
2
votes
1answer
45 views

Fake proof using mean value property

Let $f : \Delta \rightarrow \mathbb{C}$ be a holomorphic function on the unit disk. Let $u = |f|$. Assume that $u(0)=f(0)=0$. Question : since $u$ is harmonic, the mean value property should imply ...
15
votes
1answer
378 views

An outrageous way to derive a Laurent series: why does this work?

I had to compute a series expansion of $1/(e^{x}-1)$ about $x=0$, and in the course of its derivation, I made a couple of manipulations that are not allowed mathematically. Still, comparing the final ...
0
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0answers
106 views

Problem: use the well ordering principle to show that all positive rational numbers can be written in lowest terms

This problem involves pointing out the unjustified inference/logic error in the following proof that all positive rational numbers can be written in "lowest terms" that is as a ratio of positive ...
-2
votes
1answer
60 views

The problem with this simple proof [duplicate]

1) Find the error in the following proof that $2 = 1$. Consider the equation $a = b$. Multiply both sides to obtain: $a^2 = ab$. Subtract $b^2$ from both sides to get: $a^2 – b^2 = ab – b^2$. ...
4
votes
0answers
64 views

Euler's proof of divergence of sum of reciprocals of primes

On Wikipedia at link currently is: \begin{align} \ln \left( \sum_{n=1}^\infty \frac{1}{n}\right) & {} = \ln\left( \prod_p \frac{1}{1-p^{-1}}\right) = -\sum_p \ln \left( 1-\frac{1}{p}\right) \\ ...
11
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5answers
420 views

Where's the problem with a false “proof”: $\;1^0 = 1^2 \overset{?}\implies 0 = 2$

What's wrong with this: $$\large 1^0=1^2$$ Since bases are same, therefore $$\large 0=2$$ My thinking: Since the function $\,f(x)=1^x\,$ is not one to one, therefore whenever $\,f(x)=f(y),\,$ ...
1
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2answers
215 views

Why aren't all real numbers equal to one another?

I know, stupid question. But humor me for a sec. First off, we know that all real numbers have two numbers which are infinitely close to them, right? That would seem to be, for any given value of x, ...
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2answers
1k views

Lamport claims there is an error in Kelley's proof of the Schroeder-Bernstein theorem. What is it?

In section 4.1 of his note How to write a proof, Leslie Lamport mentions an error in Kelley's exposition of the Schroeder-Bernstein theorem: Some twenty years ago, I decided to write a proof of ...
4
votes
3answers
477 views

Interesting Mathematical Fallacies [duplicate]

I recently volunteered to help with a summer math program at a local high school for which I thought would be a breeze. As it turns out, it isn't a program for those catching up (summer school) like I ...
0
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2answers
124 views

Why is this wrong (complex numbers and proving 1=-1)?

$$(e^{2πi})^{1/2}=1^{1/2}$$$$(e^{πi})=1$$ $$-1=1$$ I think it is due to not taking the principle value but please can someone explain why this is wrong in detial, thanks.
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6answers
541 views

Can every indefinite integral of a discontinuous function be written in a way that “proves” something false?

I just saw the following fake proof. $$\int \frac1x dx =\int 1\cdot \frac1x dx=x\frac1x+\int x \frac1{x^2} dx = 1+ \int \frac1x dx$$ Which would imply $1=0$, hence the fake proof tag. The ...
3
votes
3answers
173 views

Validity of a trigonometric proof that $2 = 0$.

I can't find where this proof goes wrong. We know $$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A\cdot\tan B}$$ so $$\begin{align} \tan(90^\circ-45^\circ) &= \frac{\tan 90^\circ -\tan ...
1
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2answers
69 views

Simplifying $x^i$ to real numbers

I have been studying power functions, and started to think about imaginary powers. Take the function $x^i$. Because I don't know how to multiply a number $i$ times, I tried to simplify the equation ...
1
vote
1answer
94 views

What is fundamentally wrong with this?

N tosses of a fair coin. There are $\binom{2+N-1}{N} = N+1$ ways to choose with replacement from {h, t}. So the probability of N heads is $\frac 1{N+1}$. Obviously not all ways are equivalent but is ...
4
votes
1answer
444 views

Fake Proof of Prime Number Theorem

In David M. Burton's book on Elementary Number Theory I have found the following words, ... The first demonstrable progress toward comparing $\pi(x)$ with $\dfrac {x}{\ln x}$ was made by ... P. L. ...
5
votes
2answers
102 views

Why does this $u$-substitution zero out my integral?

Here's how I understand $u$-substitution working for an integral. Essentially, it involves substitution of differential expressions, allowing you to cancel out terms of the integrand. When we change ...
2
votes
2answers
86 views

Equal perimeters of squares and right angled isosceles triangles

Consider a square ABCD having length l and breadth. Now start folding the sides AB and AC so that the figure becomes something like this $$$$ All the vertical and horizontal folds/stairs are equal in ...
0
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0answers
32 views

Mistake with proof

Is there an error in the following proof? Proposition: $\forall r \in {N} ; r \neq 1$ , then $\exists$ $n \in {Z}$ such that $$2^{1/n} < r$$ Proof: Let $n$ be any integer with $$n > 1/ log_2 ...
0
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1answer
41 views

Incorrect combinatorial argument- 5 card hand with at least 3 red cards

How many 5 card hands can be made with at least three red cards? Of course, we're using a standard deck of 52. I know how to answer this, but I frequently see this argument, producing a different ...
0
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4answers
129 views

Best basic algebra examples to show students that proof by example is not sufficient [duplicate]

Often, students will try to 'prove' a propositon by checking some examples and 'concluding' that it will be true for all $n \in N$. I'm looking for some good, non-trivial examples from highschool ...
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1answer
161 views

Is $\pi = 4$ really? [duplicate]

Can anyone explain what's wrong with this?
5
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1answer
61 views

Complement of a countable open cover of the rationals

Suppose you take an open interval I of length 1, divide it into countable sub-intervals (I/2, I/4, etc.), and cover each rational with one of the sub-intervals. Since all the rationals are covered, ...
3
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2answers
73 views

Where is the error in my proof?

I have this excercise. I am able to solve it, but the problem is that I can solve it without using the last part of information of the existence of the u-vector. That makes me afraid that my proof is ...
2
votes
3answers
155 views

Proof that $\sqrt{x}=-\sqrt{x}$ [duplicate]

$\sqrt{x}=\sqrt{1\cdot x}=\sqrt{(-1)^2\cdot x} = \sqrt{(-1)^2} \cdot \sqrt{x} = (-1) \cdot \sqrt{x}=-\sqrt{x}$ The idea popped into my head while I was evaluating an integral. I have a feeling that I ...
2
votes
1answer
96 views

How to Prove it 4.1 ex.10

Prove that for any sets A, B, C, and D, if A × B and C × D are disjoint, then either A and C are disjoint or B and D are disjoint. Proof(someones). Suppose (A X B) and (C X D) are disjoint. Let (x,y) ...
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4answers
558 views

Proof of Drinker paradox [duplicate]

I searched all over the internet but didn't find a formal proof for this paradox, so here is my attempt: $\exists x[P(x)\implies \forall yP(y)]$ Let $x=x_0$. Thus $P(x_0)$ is given. Let $y$ be ...
20
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4answers
2k views

Using Integration By Parts results in 0 = 1

I've run into a strange situation while trying to apply Integration By Parts, and I can't seem to come up with an explanation. I start with the following equation: $$\int \frac{1}{f} \frac{df}{dx} ...