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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0
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3answers
42 views

Bogus set theory proof

I'm having trouble figuring out where I went wrong in this proof. I think it's to do with my understanding of things like $\cup$ and $\cap$ in that I don't really have a solid understanding of what ...
-3
votes
1answer
93 views

Does square difference prove that 1 = 2?

I was mathematically shown 1 = 2 by a function that states the following $$x^2-x^2 = x^2-x^2 $$ $$x(x-x)=(x-x)(x+x)$$ dividing by $(x-x)$ we get... $$x=x+x$$ $$ x=2x$$ $$1=2$$ I can see ...
2
votes
1answer
84 views

Mistakes in $\lim_{a\to \infty}(a^2 - a) = - \frac{1}{6}$?

One can say, using Ramanujan summation or the zeta function regularization, that the sum $\sum_{k=1}^{\infty} k=- \frac{1}{12}$. Using this result I've gotten a very confusing and counterintuitive ...
2
votes
2answers
45 views

What exactly is $\cap$-stable here?

From my lecture notes: Theorem: Let $(\Omega, \mathcal A, P)$ be a probability space, $A \in \mathcal A, \mathcal M := \{ M_1, \ldots, M_n \} \subset \mathcal A$. The following statements are ...
7
votes
3answers
275 views

What is wrong with this proof of $0=1$?

I am trying to understand what is wrong with the proof posted here that $0=1$ (source): Given any $x$, we have (by using the substitution $u=x^2/y$) ...
2
votes
0answers
27 views

$C^0([0,1])$ is separable – or isn‘t it?

Using Bernstein polynoms it can be proven that $(X, \|\cdot\|) := (C^0([0,1]), \|\cdot\|_{C^0([0,1])})$ is a seperable vector space. However, here is my “proof” that this space is not seperable: ...
2
votes
2answers
28 views

A Proof of a False Result: If $U$ is $T$-invariant, then so is $U^\perp$.

$\newcommand{\ab}[1]{\langle #1\rangle}\DeclareMathOperator{\tr}{trace}\newcommand{\mc}{\mathcal}$ I have a "proof" of the following wrong fact: Let $T$ be a linear operator on a finite ...
0
votes
1answer
60 views

Finding the error in this induction proof [duplicate]

Claim: If $n$ belongs to $\mathbb{N}$, and $p$ and $q$ are natural numbers with maximum $n$, then $p=q$. Let $S$ be the subset of the natural numbers for which the claim is true. $1$ belongs to $S$, ...
7
votes
2answers
270 views

Was Smullyan really wrong?

EDIT: the OP has since edited the question fixing all the issues mentioned here. Yay! There was a question asked on Puzzling recently, titled ...
1
vote
1answer
28 views

Error in proof: Distribution of exponents for negative number [duplicate]

Here are steps of the "proof": $1=1$ $\Rightarrow 1=\sqrt{1}$ $\Rightarrow 1=\sqrt{-1\times-1}$ $\Rightarrow 1=\sqrt{-1}\times\sqrt{-1}$ $\Rightarrow 1=i\times i$ $\Rightarrow 1=-1$ At which ...
3
votes
4answers
128 views

Where does this argument showing there are uncountably many TMs fail?

This argument comes up once every while on Lambda the Ultimate. I want to know where the flaw is. Take a countable number of TMs all generating different bitstreams. Construct a Cantor TM which runs ...
-1
votes
2answers
52 views

How can the ball reach the wall when it always has to travel halfway? [duplicate]

If I throw a ball at the wall, when it has travelled halfway, it still has half the distance to travel. As it continues, the fraction left to travel continues i.e. one quarter to go, one eighth to go, ...
1
vote
2answers
48 views

Is $\sum_{n=1}^\infty a_n\sin(nx)$ converges on $[\varepsilon, 2\pi-\varepsilon]$?

Let $a_n$, a sequence monotonically decreasing to $0$. Consider $$\sum_{n=1}^\infty a_n\sin(nx)$$ Is the series converges uniformly on $[\varepsilon, 2\pi-\varepsilon]$? ($\varepsilon ...
1
vote
2answers
56 views

Do we really need the constraint qualification?

I can't keep my fingers off Nocedal/Wright's Numerical Optimization (1999,1E) and I apologize. But maybe YOU can shed light on the question: Why does a point $x \in \mathbb{R}^n$ need to satisfy the ...
-3
votes
1answer
41 views

Is this mathematical statement? [closed]

$\{\text{integers $n$ such that $n$ is even}\}$ It can be true/false so does that mean it's proposition/mathematical statement?
1
vote
1answer
57 views

Where is the problem here:$-1=(-1)^1=(-1)^\frac{2}{2}=({(-1)}^{2})^{1/2}=\sqrt{1}=1$? [duplicate]

Is there someone show me Why this is not true ? $$-1=(-1)^1=(-1)^\frac{2}{2}=({(-1)}^{2})^{1/2}=\sqrt{1}=1$$ then :$$-1=1$$ Thank you for any help
8
votes
6answers
196 views

I was wondering, shouldn't the fraction $\frac {-2}{-1}$ be less than 1?

Because technically, the numerator is smaller than the denominator as $-2 < -1$ I know it's an extremely stupid question. I mean I know that I can just multiply $-1$ to the numerator and the ...
0
votes
2answers
63 views

Paradox - minus one equals one using square roots [duplicate]

I was looking on Howard Eves's book "An Introduction to the History of Mathematics" and I stumbled upon a demonstration on how $-1 = 1$. The demonstration follows: $$ \sqrt{-1} = \sqrt{-1} $$ $$ ...
0
votes
1answer
31 views

P(TT|T) in two coin tosses not 1/3?

So the question is, if two consequtive coin tosses occured and we know, in the aftermath, that at least of them resulted in a tails, what is the probability that they were both tails? The common ...
0
votes
3answers
86 views

What's wrong with this?

What's wrong with this : $$e^{i\pi} = -1$$ $$\therefore e^{2i\pi} = 1$$ $$\therefore log \left( e^{2i\pi} \right ) = log(1) = 0$$ $$\therefore 2i\pi = 0$$
0
votes
1answer
16 views

Using Union to prove a context-free language? [closed]

I am working through many examples and I seem to have confused myself and made all the questions rather trivial. If I have the CFLs, $L_1 = \{1^n 0^{mn} : n,m \in \Bbb N\}$ and $L_2 = \{1^m 0^n : ...
4
votes
1answer
112 views

What is wrong with this “proof” that there is no $\omega$th inaccessible cardinal?

"Theorem": There is no $\omega$th inaccessible cardinal. "Proof": Assume ZFC. Let $\kappa_n$ be the $n$-th inaccessible cardinal; since $V_\kappa$ is a model of ZFC for inaccessible $\kappa$, ...
0
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1answer
75 views

Mathematical Induction. Horses made me question my understanding

I recently read about the false inductive proof that all horses are the same colour. There are some mathSE threads about this already (MathSE_thread_1, MathSE_thread_2). After reading this, I now ...
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votes
4answers
170 views

Why is this proof of $i = -1$ wrong? [closed]

I know that $-7^2 = -49$ Therefore $\sqrt{-49} = -7$ Because $\sqrt{-1} = i$ we can then expand it to $\sqrt{-49} = -7 = 7i$ And therefore $-7 = 7i$, divide both sides by 7 and you get $-1 = i$ ...
0
votes
1answer
40 views

Check my logical argument for this proof.

if x is a real number $x \not =\ 1 $, then there exists y which is also a real number $ ((y+1) \div ( y-2) ) = x .$ Prove it's converse also. Logical Argument: given: $x \not = 1$ Goal: $ ...
6
votes
3answers
198 views

Where does the proof of $\sqrt 2$ is irrational break down when trying to prove the same for $\sqrt 4$?

I'm trying find where the common proof by contradiction that $\sqrt 2$ is irrational breaks down when trying to prove $\sqrt 4$ is irrational. Assume $\left(\frac pq\right)^2=4$ and $\gcd(p,q)=1$. I ...
4
votes
5answers
185 views

Is $\sqrt{x^2} = (\sqrt x)^2$? [duplicate]

Take $x=4$ for example: $ \sqrt{(4)^2} = \sqrt{16} = \pm4 $ However: $ (\sqrt{4})^2 = \sqrt{\pm2}$ Case 1: $ (-2)^2 = 4$ Case 2: $ (2)^2 = 4$ Solution : $+4$ How come the $ \sqrt{(4)^2} = \pm4$; ...
2
votes
1answer
50 views

Same Expansion for Different Functions! What's Wrong?

We know, the function $arctan(x)$ is defined as the real number $y$ such that $tan(y)=x$ and $-\frac{\pi}{2}\leq y\leq\frac{\pi}{2}$ From this definition we can derive the well-known ...
2
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0answers
264 views

An argument for “Brocard's problem has finite solution”

Brocard's problem is a problem in mathematics that asks to find integer values of n for which $$x^{2}-1=n!$$ http://en.wikipedia.org/wiki/Brocard%27s_problem. According to Brocard's problem ...
-3
votes
3answers
191 views

$1+1=0$ What am I doing wrong???! [duplicate]

Does someone know what I'm doing wrong? I'm struggling with this for a while now and I don't see what I do wrong! $$1+1=$$ $$1+\sqrt{1}=$$ $$1+\sqrt{-1*-1}=$$ $$1+\sqrt{-1}*\sqrt{-1}=$$ $$1+i*i=$$ ...
4
votes
1answer
76 views

Fake proof: Equivalence of norms

Good morning. I'm having a hard time finding what's wrong with the following argument. Let $f$ be any function in $C^{1}([0;1])$ and let $||f||$ and $N(f)$ be two norms defined as follows: $$||f|| = ...
15
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7answers
2k views

What is wrong with this putative proof?

So I've spent about an hour trying to figure out what is wrong with this proof. Could somebody clearly explain it to me? I don't need a counterexample. For some reason I was able to figure that out. ...
1
vote
0answers
43 views

Proof subtraction is not forward stable

I've been taught that the "subtraction operation" is not accurate/forward stable as the relative error can be arbitrary large. I tried to prove it formally but I end up with a contradiction. What ...
3
votes
1answer
100 views

What is wrong with this proof that $\text{card}(\mathcal{P}(\mathbb{N}))=\aleph_1$?

I'm reading books on set theory and I came up with the following 'proof' that $\text{card}(\mathcal{P}(\mathbb{N}))=\aleph_1$. What is wrong with it? I really cannot tell! (By ...
1
vote
1answer
36 views

Is the set of vanishing $k$ derivatives of smooth functions in a null set dense in $W^{1,p}$?

Let $\Omega$ be an open set with compact closure denoted by $\overline{\Omega}$ and a null set $N\subset\Omega$ with respect the Lebesgue measure. Then consider the two sets ...
46
votes
10answers
1k views

Fake induction proofs

Question: Can you provide an example of a claim where the base case holds but there is a subtle flaw in the inductive step that leads to a fake proof of a clearly erroneous result? [Note: Please do ...
0
votes
3answers
58 views

E is countable $\longleftrightarrow$ there exist a surjection from $\mathbb{N}$ to E

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. I was given the following problem: Prove that a set $E$ is countable if and only if there ...
4
votes
4answers
1k views

Why is “All horses have the same color” considered a false proof by induction? [duplicate]

Upon reading of All horses have the same color "paradox", I began to wonder a couple of things. First of all, to me the inductive step seems flawed. Just because I have $n$ white horses, does not ...
12
votes
6answers
796 views

Induction - Examples where the induction step is correct but the base case is always wrong [duplicate]

I'd like to present to my students some induction examples that always satisfy the inductive step but never the base case. It could be for natural numbers, graphs or anything else.
1
vote
3answers
139 views

Why is $(-1)^3=(-1)^{6/2}=((-1)^6)^{1/2}=1^{1/2}=1$ wrong? [duplicate]

Why is this wrong? $$(-1)^3=(-1)^{6/2}=((-1)^6)^{1/2}=1^{1/2}=1$$ It seems logical but I know it's wrong.
1
vote
1answer
59 views

Explain what is wrong with the following “proof” by induction.

Basically, there's so much going on in this problem that I don't even understand it. I've read it about one million times, but it still isn't making sense to me. Any hints would be appreciated... ...
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votes
1answer
104 views

Did I just prove 1 = 0 [duplicate]

Consider the formula for integration by parts $\int u dv = uv - \int v du$ Now let's apply an operator $\frac{\partial^2}{\partial u\partial v}$ on both sides: $\frac{\partial^2}{\partial u\partial ...
0
votes
0answers
19 views

Fallacious proof with induction [duplicate]

my teacher gave the following as an example of a fallacious proof: We'll prove that a group of $n$ people are either all male or all female. For $n = 1$ The claim says that a group containing ...
1
vote
1answer
76 views

Counterexample to Schwarz Lemma

Schwarz Lemma states the following: Let $D = \{z : |z| < 1\}$ be the open unit disk in the complex plane centered at the origin and let $f : D \to D$ be a holomorphic map such that $f(0) = 0$. ...
3
votes
3answers
70 views

Spot mistake in finding $\lim \limits_{x\to1}\left(\frac x {x-1} - \frac1 {\log(x)} \right)$

This is the limit I'm trying to solve: $\lim \limits_{x\to1}\left(\frac x {x-1} - \frac1 {\log(x)} \right)$ I thought: let's define $x=k+1$, so that $k\to0$ as $x\to1$. Then it becomes: $$\lim ...
3
votes
1answer
104 views

Euler Mascheroni Constant is Zero

$$ H_n - \ln n = \int_0^1 \frac{1 - x^n}{1 - x} dx - \int_1^n \frac{dy}{y} $$ Let $x = \frac{y - 1}{n - 1}$, or $y = (n-1)x + 1$. Then, $dy = (n - 1) dx$. $$ H_n - \ln n = \int_0^1 \frac{1 - x^n}{1 ...
-4
votes
2answers
71 views

how can be this possible? What is wrong with this. [duplicate]

We can see that 1^2 =1 ; 2^2 =2+2 ; 3^2=3+3+3 ; . . . x^2=x+x+x+..... (x times) differentiation on both sides gives 2x=1+1+1+....... (x times) 2x=x What's happening hear.How is this possible. ...
-2
votes
5answers
153 views

Fake proof $2=1$ [closed]

let $$x=y \implies 2x-x=2y-y \implies 2x-2y=x-y$$ $$2(x-y)=(x-y) \implies 2=1 \ \ \ \ \operatorname{By Cancellation Law}$$
2
votes
1answer
90 views

What is wrong with $\sqrt{-1} = (-1)^{1/2} = (-1)^{2 \times {1/4}} = (-1^2)^{1/4} = 1^{1/4} = 1$?

Why $\sqrt{-1} = (-1)^{1/2} = (-1)^{2 \times {1/4}} = (-1^2)^{1/4} = 1^{1/4} = 1$ is not true?
2
votes
3answers
74 views

Help Figuring Out Faulty Proof

In my discrete math class, we're working on faulty proofs. I can't seem to figure out why this proof is faulty. I think it has to due with them assuming $k^2 \le k^2 + 2k$. Anyone have any ideas? ...