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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5
votes
3answers
93 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 $(\frac pq)^2=4$ and $\gcd(p,q)=1$. I guess I ...
4
votes
5answers
160 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
42 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 ...
3
votes
0answers
150 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
146 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
65 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|| = ...
13
votes
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
36 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
93 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
31 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 ...
38
votes
9answers
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
50 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
688 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
130 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
52 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... ...
-4
votes
1answer
92 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
59 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
64 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
97 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
67 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
147 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
89 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
63 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? ...
2
votes
1answer
74 views

Show $\left|{\frac{z_1-z_2}{1-z_1 \overline{z_2}}}\right| < 1$ if $|z_1| <1$ and $|z_2| < 1$

Show $$\left|{\frac{z_1-z_2}{1-z_1 \overline{z_2}}}\right| < 1$$ if $|z_1| <1$ and $|z_2| < 1$ Consider: $$\left|{\frac{z_1-z_2}{1-z_1 \overline{z_2}}}\right|^2$$ ...
1
vote
0answers
53 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
vote
0answers
32 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$
-3
votes
2answers
101 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
votes
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
vote
0answers
38 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
votes
1answer
15 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
votes
1answer
45 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} ...
2
votes
1answer
178 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
votes
4answers
199 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
votes
3answers
68 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 ...
0
votes
2answers
60 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 $$-3x^2+7 = -3y^2+7$$ $$-3x^2 = -3y^2$$ $$x^2 = y^2$$ $$x = y$$ ...
8
votes
2answers
145 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 ...
2
votes
6answers
173 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
votes
1answer
64 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 ...
0
votes
2answers
95 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
570 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
124 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
votes
1answer
78 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 $\sqrt{-1}$? $-1 = i\cdot i = \sqrt{-1}\cdot \sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{1} = 1$
1
vote
0answers
28 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 ...
0
votes
1answer
22 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
votes
3answers
180 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
votes
2answers
52 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
138 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
vote
0answers
149 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 ...