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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-3
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1answer
36 views

Is this mathematical statement? [on hold]

$\{\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
48 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
7
votes
6answers
179 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
51 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
24 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
75 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
14 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
107 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
votes
1answer
69 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 ...
-2
votes
4answers
164 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
38 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
185 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
180 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
48 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
220 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
180 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
74 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
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
41 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
98 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
34 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 ...
42
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
56 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
750 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
58 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
100 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
71 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
103 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
151 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
70 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$
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votes
2answers
108 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
40 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
17 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
51 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
183 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
205 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
71 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
63 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
152 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
187 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*} ...