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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21
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7answers
3k views

What is the flaw of this proof (largest integer)?

Let $n$ be the largest positive integer. Since $n ≥ 1$, multiplying both sides by $n$ implies that $n^2 ≥ n$. But since $n$ is the biggest positive integer, it is also true that $n^2 ≤ n$. It follows ...
0
votes
1answer
27 views

Covariance of dice tosses that result in 1 or 2 (fake proof)

Question: Consider n independent tosses of a $k$-sided fair dice. Let $X_i$ be the number of tosses that result in $i$. What is the covariance $\mathrm{cov}(X_1,X_2)$ of $X_1$ and $X_2$. ...
0
votes
1answer
40 views

Requirements too stringent for singleton homotopy class [X,Y]?

I recently had a problem: Show that if $X$ is contractible, and $Y$ is path-connected, show that the homotopy class $[X,Y]$ has a single element. I have been able to prove this (I think) in a fairly ...
1
vote
1answer
40 views

Error with the proof that all solutions to the Cauchy Functional Equation are linear

If $f(x)$ is continuous, it is known that $f(x+y)=f(x)+f(y)$ implies that $f(x)$ is linear, and non-continuous solutions are discussed in these links. (1, 2,3, 4) However, what is wrong with this ...
10
votes
4answers
200 views

Why isn't $e^{2\pi xi}=1$ true for all $x$?

We know that $$e^{\pi i}+1=0$$and $$e^{\pi i}=-1$$ So$$(e^{\pi i})^2=(-1)^2$$$$e^{2\pi i}=1$$ Because $1$ is the multiplicative identity,$$(e^{2\pi i})^x=1^x$$$$e^{2\pi xi} =1$$should also hold ...
0
votes
1answer
189 views

Prove 1 is not the largest integer? [duplicate]

This proof looks extremely flawed, but I'm new to proofs so I'm not completely sure what is allowed and what isn't. Here it is: Let $n$ be the largest positive integer. Then $n$ must be $\geq 1$. ...
1
vote
2answers
40 views

Finding error in an incorrect proof

Statement: If $a$, $b$ and $b'$ are integers and $a>b>b'>0$, then the remainder when $a$ is divided by $b$ is less than the remainder when a is divided by $b'$. Proof: Assume $a,b, b'$ are ...
0
votes
1answer
34 views

How do you define such map $(C^B \times B^A) \to C^A$?

Suppose that $\mathbf{C}$ be cartesian closed and $B$ is an object of it. We define two functors $\mathbf{C} \times \mathbf{C} \to \mathbf{C}$ by $$ C^B \times B^A \qquad\text{and}\qquad ...
0
votes
0answers
32 views

Proof of a series law

I'm stuck on the following exercise: "Let $\sum_{n=m}^\infty a_n$ be a series of real numbers, and let $k\geq 0$ be an integer. If one of the two series $\sum_{n=m}^\infty a_n$ and ...
0
votes
2answers
63 views

Is there any flaw in the proof of $-1 = 1$? [duplicate]

We have $i^2 = -1$ where $i = \sqrt{-1}$. Now consider $$\sqrt{-1}\cdot \sqrt{-1} = \sqrt{(-1)\cdot(-1)} = \sqrt1 = 1$$ Which proves $-1 = 1$. Is there anything wrong with ...
23
votes
10answers
662 views

What are the Laws of Rational Exponents?

On Math SE, I've seen several questions which relate to the following. By abusing the laws of exponents for rational exponents, one can come up with any number of apparent paradoxes, in which a number ...
0
votes
2answers
78 views

1 = -1 Clearest way to explain why this proof is wrong. [duplicate]

Say you are a high school student or a young undergrad. You are being taught about complex numbers and you are told that $i = \sqrt{-1}$. You go home and you write this: \begin{equation} ...
3
votes
1answer
106 views

$\sqrt{x}$ is a constant function?

I just "proved" something ridiculous and can't find the fault in my logic. It's probably something really simple and obvious that I'm just overlooking, or maybe not because none of my friends can find ...
1
vote
2answers
52 views

Proof: divisibility

Question: For all $a, b, c \in \mathbb{Z}$, if $a\mid bc$, then $a\mid b$ or $a\mid c$. Is this true? My answer: True. (Proof by contrapositive) Proof that if $a \nmid b$ and $a \nmid c$, then $a ...
0
votes
1answer
28 views

In the following equation $Av = \lambda v$ $v$ cannot be $\vec{0}$

In this question and answers to such question people say that $v$ cannot be $\vec{0}$, but this not correct in my opinion. We assume it is different from $\vec{0}$ because otherwise $Av = \lambda v$ ...
-1
votes
3answers
47 views

Why my answer for this conditional probability problem is wrong? [duplicate]

A family has two children. What is the probability that both the children are boys given that at least on of them is a boy? Solution given in my book is My doubts and my solution If a family has ...
1
vote
3answers
108 views

Dilemma about the value of $\frac{4- 4}{4 - 4}$

I can't find where the mistake is here. Can someone explain how it is possible?
1
vote
2answers
104 views

A “bijection” from $\mathbb{N}$ to $\mathbb{R}$?

We know that there doesn't exist a bijection from $\mathbb{N}$ to $\mathbb{R}$. But the following 'argument' proposes to find such an "bijection". First Fallacious Argument Let $a_1\in ...
0
votes
0answers
36 views

Computing the ramification index of a morphism of curves

Definition: Let $f: C_1 \to C_2$ be a nonconstant map of smooth curves and let $P \in C_1$. $$e_f (P) = \textrm{ord}_P (f^* t_{f(P)})$$ where $t_{f(P)} \in K(C_2)$ is a uniformizer at $f(P)$ ...
0
votes
2answers
84 views

What's wrong with this natural deduction proof?

According to natural deduction $\forall$ $x$ $\exists$ y $P(x,y)$ $\models$ $\exists$ $x$ $\forall$ y $P(x,y)$ is incorrect. However I am able to prove the following using the rules of natural ...
3
votes
0answers
46 views

Global sections of Proj

In constructing the relative Proj for a graded ring $S$, one inevitably has that the distinguished opens, $D_+ (f)$, where $f \in S_+$ gives rise to a scheme that is isomorphic to $\textrm{Spec } ...
1
vote
6answers
137 views

“Proving” that $0^0 = 1$ [duplicate]

I know that $0^0$ is one of the seven common indeterminate forms of limits, and I found on wikipedia two very simple examples in which one limit equates to 1, and the other to 0. I also saw here: ...
-4
votes
2answers
109 views

Where does this go wrong? (7=5) [duplicate]

Show me my mistake: $$a=b\Longleftrightarrow$$ $$a^2=ab\Longleftrightarrow$$ $$7a^2=5ab+2a^2\Longleftrightarrow$$ $$7a^2-7ab=5a^2+2ab-7ab\Longleftrightarrow$$ $$7a^2-7ab=5a^2-5ab\Longleftrightarrow$$ ...
0
votes
1answer
57 views

Is this a tautological argument?

Assume $A$ is true, and that $A$ implies $B$. If I then can prove that if $A$ implies $B$, then $B$ implies $A$, have I then proved that $A$ is true (without the need to assume that $A$ is true in the ...
9
votes
4answers
143 views

What does the false infinite sum of a series mean?

For any geometric series with |$r$| < 1 , I know that $$\sum_{k=1}^{∞} ar^{k-1} =\frac{a}{1-r}$$ But if |$r$| > 1 and you try to use the formula, you'll get a weird answer. For instance: ...
0
votes
4answers
79 views

Why is this proof considered wrong? [closed]

I was asked to prove following statement: $$\log_a{(x_1\cdot x_2)} = \log_a{(x_1)} + \log_a{(x_2)}$$ What I did was: \begin{align} \log_a{(x_1\cdot x_2)} &= \log_a{(x_1)} - (-1)\cdot ...
-3
votes
2answers
64 views

Disprove why 0 ∉ Z [closed]

Disprove this A value x is said to be an integer when floor(x) = x, where x ∈ ℝ floor(x)/x = 1 Therefore floor(x)/x ∈ Z, where x ∈ ℝ And since 0 ∈ ℝ From the definition of an integer, ...
0
votes
2answers
55 views

Volumes of solid of revolution: sin(x) + 2 fake proof

I just recently learned about volumes of solids of revolution in my AP Calculus class and tried to create a problem to connect it to related rates. In this process, I found an error that neither my ...
2
votes
1answer
46 views

Show that if $\angle ADB = 60^{\circ}$ then $AA_1 = BB_1$ (and answer whether the converse is true).

In the diagram above we have that $AA_1$ and $BB_1$ are altitudes and $\angle ADB = 60^{\circ}$. The problem is two fold- show that from$\angle ADB = 60^{\circ}$ it follows that $AA_1$ = $BB_1$ and ...
3
votes
4answers
233 views

Mathematical fallacy of $x^{x^{x^{x^x…}}}$ = 2

Suppose we have an equation with an infinite number of $x$'s as an exponent: $$x^{x^{x^{x^x...}}} = 2$$ $$x^{(x^{x^{x^x...}})} = 2$$ because there are infinity $x$'s in the parentheses, which we've ...
3
votes
1answer
94 views

Fake proof that there don't exist complicated numbers

So there's this false proof going around that I can't seem to find now that says that complicated numbers don't exist. So let me explain what it's about (I've added some technical details of my own, ...
1
vote
2answers
59 views

Prove that for all n>11 we can represent $n$ by $n=3a+7b$

So i decided to do this using normal induction. $P(12)$ true since $12=4 \times 3$ $P(k)=3a+7b$ $P(k+1)=3(a-2)+7(b+1)$ So i think it is proven but i cannot see why we have to assume $n$ is larger ...
47
votes
15answers
15k views

Zero divided by zero must be equal to zero

What is wrong with the following argument (if you don't involve ring theory)? Proposition 1: $\frac{0}{0} = 0$ Proof: Suppose that $\frac{0}{0}$ is not equal to $0$ $\frac{0}{0}$ is not equal to $0 ...
13
votes
4answers
2k views

What is the fallacy in this proof? [duplicate]

I came across this funny proof- $$4$$ $$=4+\frac 92-\frac 92$$ $$=\sqrt{(4-\frac 92)^2}+\frac 92$$ $$=\sqrt{16+\frac{81}{4}-36}+\frac 92$$ $$=\sqrt{25+\frac {81}{4}-45}+\frac ...
3
votes
0answers
53 views

Example of Skew-Symmetry of Connection Forms

As is commonly known, the connection 1-forms of a Riemannian manifold are skew-symmetric: $\omega^i_j=-\omega^j_i$. Until now, I have not actually thought to hard on this, but I think I've hit a snag. ...
1
vote
1answer
41 views

Length of diagonal compared to the limit of lengths of stair-shaped curves converging to it [duplicate]

I see this post and I am stunned. I think this is fallacious but I can't figure where is the fallacy? If you know the fallacy. Please post a answer.
2
votes
4answers
92 views

Fake Proof: $a^n = 1$ f or all nonnegative integers $n$

Find the flaw with the following "proof" that $a^n = 1$ for all nonnegative integers $n$, whenever $a$ is a nonzero real number. Basis Step: $a^0 = 1$ is true by definitino of $a^0$. ...
2
votes
4answers
124 views

Is $i$ equal to $-i$? [duplicate]

When I was in high school, I learned about $i$ in math class and I remember asking my teacher back then if $i$ was equal to $-i$ according to the simple following development : \begin{equation} ...
0
votes
1answer
81 views

Stumbled on a “proof” to generate primes and I cannot find the wrong step.

Let $P_k$ be a prime number and let $P = 2.3.\dots.P_k$. be the product of all primes smaller or equal to $P_k$. Then $P+1$ is either a prime number or not. If it is not a prime number it shares ...
4
votes
2answers
97 views

A paradox I came up with, proof verification.

I am wondering if the following is right? This is my solution to the following paradox.$$R = \{x : x \text{ not within }x,\,\text{where }x\text{ is a set}\}.$$Since by the nature of Russell's ...
1
vote
1answer
59 views

What's wrong here? [duplicate]

$$i=\sqrt{-1}=\sqrt{\frac{1}{-1}}=\frac{1}{i}=\frac{-i^2}{i}=-i$$ I'm sure there is a mistake above but I can't figure out where. What's exactly wrong in the above situation & why?
2
votes
2answers
33 views

If $f:X\to Y$ is linear and $y^\ast \circ \ f \in X^\ast$ for all $y^\ast \in Y^\ast$, then $f$ is continuous

Let $X,Y$ be normed over $\mathbb{K}$ (which can be $\mathbb{R}$ or $\mathbb{C}$). If $f:X\to Y$ is linear and $y^\ast \circ \ f \in X^\ast$ for all $y^\ast \in Y^\ast$, then $f$ is continuous. I ...
0
votes
1answer
45 views

Flaw in proof involving binomial theorem

Suppose we have $-1 < x < 0$, also an irrational $r$. We have three claims: $x^r$ is not defined on $\mathbb{R}$(pretty obvious right) $(1+x)^r$ is defined on $\mathbb{R}$(once again, pretty ...
1
vote
1answer
73 views

Strange problem with the imaginary unit [duplicate]

In class while messing with fractions and complex numbers I found this "paradox" $$ \sqrt{-1}=\sqrt{-1} $$ $$ \sqrt{\frac{-1}{1}}=\sqrt{\frac{1}{-1}} $$ $$ ...
0
votes
1answer
42 views

Math: amounts and a proof about amounts

I would like to proof or disprove the following 2 statements, if any is not true I need to find an example which disproves it. X and Y are amounts and f: X --> Y This is a question where I have ...
-3
votes
4answers
123 views

What is wrong with the following proof that tries to prove that $2=1$? [duplicate]

I need to find the flaw in the following proof: $a,b\in\mathbb{R}$\ $\left\{ 0 \right\} $ such that $a=b$ 1) Multiplying both sides by $a$ yields the equality: $a^2=ab$ 2) Subtracting $b^2$ from ...
0
votes
2answers
55 views

Is there a way to prove this wrong?

We have two statements: 0 = 1 Both statements are false If the second statement is true, then it is false. We've come to a contradiction. If the second statement is false, then at least one of ...
0
votes
1answer
22 views

Do $L^{2}$ energy estimates implies $H^{k}$ weak solutions?

The answer I think must be no. However I can not see what is wrong with the following reasoning. Assume we have an estimate of the form for the adjoint of an operator $L$ in $L^{2}$ of the form: ...
2
votes
1answer
38 views

$\lim\limits_{(x,y) \to (0,0)} \frac{{x{y^2}}}{{{x^2} + {y^4}}} = 0$ prove by polar coordinates.

so we have I have the limit $\lim\limits_{(x,y) \to (0,0)} \frac{{x{y^2}}}{{{x^2} + {y^4}}}$ Now i know that this limit does not exist. I even saw the proof on this website. However, if you substitute ...
10
votes
15answers
572 views

$2=1$ Paradoxes repository

I really like to use paradoxes in my math classes, in order to awaken the interest of my students. Concretely, these last weeks I am proposing paradoxes that achieve the conclusion that 2=1. After one ...