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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22
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3answers
2k views

Where does the gap come from? [duplicate]

Can anyone tell me please where does the gap come from? Thanks and sorry if the question is not exactly relevant, I just didn't know where else to ask.
-8
votes
3answers
126 views

Please attempt to fault my proof of the continuum hypothesis [on hold]

I have put this proof of the continuum hypotheses to both a Dr. of Maths and a Professor of Logic, and neither has demonstrated a flaw - although I doubt the professor (who shall remain nameless) ...
0
votes
1answer
55 views

Direct proof that if mn is odd then m is odd and n is odd

I found the converse here, although that's not what I want. I have thought of a proof by contradiction and by contraposition, although I can't seem to figure out a way to finish a direct proof. $mn ...
0
votes
1answer
31 views

Whats wrong with this proof? (infinite sequences) [duplicate]

So I just watched a video where they explained that the sum of all natural numbers is $-1/12$. However, there was an interesting comment: ...
0
votes
5answers
85 views

Proof that $\{n\}$ is a Cauchy Sequence. Where is the fallacy?

We need to show that for every $\varepsilon>0$, $\exists N \in \mathbb R$ such that $n,m > N \implies |n-m|<\varepsilon$. $|n-m|<n+m$. So, if we make $n+m< \varepsilon$, the result ...
1
vote
3answers
112 views

Why does the same equation have different results?

I bring the equation in (1) in order to ilustrate what I mean. Since (1.1) $12 - 6$ (1.2) $(4*3) - (2*3)$ (1.3) $(4-2) * (3)$ (1.4) $(2) * (3)$ (1.5) $6$ So... (2.1) $9.999... - 0.999...$ ...
1
vote
0answers
70 views

Question on an April Fool on Fourier transform

I have a question on this answer : Let $f(x) = 1$. It's easy to see that its Fourier transform is $0$ almost everywhere, so $\hat {\hat f}(x) = 0$. By the inversion theorem, $1 = 0$. I think ...
21
votes
6answers
4k views

Understanding Euclid's proof that the number of primes is infinite. [duplicate]

In Euclid's proof, if $p_1, p_2, \dots, p_n$ are the only primes then $p_1 \times p_2 \times \dots \times p_n + 1$ is not divisible by any of $p_1, p_2, \dots, p_n$ (because of some algebraic facts), ...
2
votes
1answer
68 views

Prove that if $A$ and $B$ are square matrices and $AB$ is invertible, then both $A$ and $B$ are invertible

I already know how to prove this using the definition of inverse and the associative property of matrix multiplication, but I was wondering if this would also be a valid proof. As $A$ and$ B$ are $n ...
4
votes
1answer
901 views

April Fools' Day Hoax: Fermat's Last Theorem

I read this answer to this question on MathOverflow, and I enjoyed reading the proof given in the linked paper, but... where is the mistake? I know nothing of the Mason-Stothers Theorem except its ...
0
votes
3answers
23 views

Find the Fallacy in the Factor Group Logic

The factor group for $(\Bbb{Z} \times \Bbb{Z})/\langle(2, 0)\rangle$ is clearly $\Bbb{Z} \times \Bbb{Z}_2$ because all of the elements are in the form of $a(1, 0)+b(0, 1)+\langle(2, 0)\rangle$ for $a ...
4
votes
2answers
121 views

What exactly is the 'induction trap'

I've looked everywhere, and I've looked at a lot of examples. I don't quite understand what about the induction trap is so wrong. The most common example is the graph theory tree example (page 5 here: ...
0
votes
0answers
12 views

Semigroup of a common face of two cones in a fan

Let $\Sigma$ be a fan, $\sigma_1, \sigma_2 \in \Sigma$, and $\tau = \sigma_1 \cap \sigma_2$ be a common face of the two cones. For a cone $\sigma$, denote the semigroup by $S_\sigma = \sigma ^\vee ...
1
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1answer
65 views

What is the proof for $\sqrt{-a}\times\sqrt{-b}\neq\sqrt{ab},\text{ where }a,b\in \mathbb{R}$

Having just learned about $i$ in my 10$^{th}$ grade classroom I'm interested in the proofs underlying the rules for algebraic manipulations with imaginary numbers; such an understanding will create a ...
1
vote
1answer
55 views

Fundamental theorem of Algebra in Complex Plane

Edit: I know that Euler had a method, which was incomplete, then Gauss proved the FTA. If somebody could show me what Euler did, it would be great. I think that's what I need. The Aim is to prove the ...
2
votes
0answers
33 views

Why do the integral and the partial sum agree for small $a$ and $m$?

Consider the following naive manipulations: \begin{align} \int_0^\infty \frac{e^{-x}}{1+ax}\:dx & = \int_0^\infty e^{-x}\frac{1}{1-(-ax)}\:dx\\ &= \int_0^\infty e^{-x} \left( ...
6
votes
4answers
147 views

Just got confused with what my friend asked (paradox and fake proofs). [duplicate]

Take $x^2=x+x+x+\cdots$ ($x$ times). Now differentiating both sides wrt $x$, we get: $$2x=x.$$ This means $x=0$ or $2=1$. How? Where did I go wrong?
1
vote
1answer
127 views

The three-coin-flip riddle

Is the following true (It seems obvious to me that it's not... but... a PhD in physics, Derek Abbott, seems to think others explanation at end of post): Someone flips 3 coins on the table, they are ...
0
votes
1answer
23 views

Birkhoff Ergodic theorem for two measures

Suppose $(X,\mathcal{B}, \mu, T)$ and $(X,\mathcal{B}, \nu, T)$ are both ergodic ppt. I'm a bit confused how the B Ergodic Theroem works since the LHS of the equation doesn't involve $\mu$ or $\nu$, ...
0
votes
1answer
84 views

Proving Infinite Nested Radical

This question asks to prove the limit of the infinitely nested radical. Now, I only have vague idea of what rigor means in proving something, but seeing my "answer" being radically different from ...
0
votes
1answer
56 views

How do I demonstrate that a logical error has been made?

I am grading an introduction to proofs course at my university. We have a question that reads something along the lines of: Prove that $$\forall \ n \in \Bbb N, [\exists \ a \in \Bbb N : n + 1 = 3a ...
9
votes
5answers
189 views

Infinite summation: $x+x+x+x+… =2$?

One of my favourite little math problems is this $x^{x^{x^{x^{...}}}}=2$ The solution to it is quite simple. An infinite tower of x's is equal to 2, and above the first x there is still an infinite ...
32
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3answers
3k views

Can't find the flaw in the reasoning for this proof by induction?

I was looking over this problem and I'm not sure what's wrong with this proof by induction. Here is the question: Find the flaw in this induction proof. Claim $3n=0$ for all $n\ge 0$. ...
-2
votes
1answer
37 views

What is wrong with this reasoning when calculating circle perimeter? [duplicate]

Looking at the following image, which was posted on the internet: Could someone tell me what is wrong? It seems true for the first 4 small images. But, when it comes to infinitesimal length, ...
0
votes
1answer
48 views

Contradiction in value of $\psi (1)$

What is the value of $\psi (1)$ ? If we take the definition in terms of derivative of Gamma function, we get $\psi (1) = \dfrac{\Gamma'(1)}{\Gamma(1)} = -\gamma$. But, if we consider the series ...
2
votes
1answer
64 views

What is wrong with the given proof? [duplicate]

Here is the proof they gave: Start with the statement $a = b$. Multiply both sides by $b$ to get $ab = b²$. Subtract $a²$ from both sides to get $ab − a² = b² − a²$. Factor the left and right sides of ...
0
votes
2answers
79 views

Alternative triangle inequality proof

I have looked everywhere for confirmation of this proof of the triangle inequality with no success. Prove the triangle inequality: $$\vert x + y \vert \leq \vert x \vert + \vert y \vert.$$ Proof: ...
0
votes
2answers
123 views

Another $1=2$ proof [duplicate]

So a friend shows me this : $x^4= x^2+x^2+ \cdots +x^2 $ ( i.e. $x^2$ added $x^2$ times) Now take the derivative of both side; $4x^3 = 2x + 2x + \cdots + 2x $; So $4x^3 = 2x^3 \cdots $(1) And ...
2
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4answers
51 views

Proof verification : every algebraic set is the union of finitely many irreducible algebraic subsets

I have found various proofs of the result but I have come up with something very different and I wonder whether it is a valid argument: Let $W$ be an algebraic set. Let $I=\mathcal{I}(W)$. We have ...
0
votes
2answers
33 views

Vacuous statements and explosion

So my understanding of vacuous statements is as follows: For any statement $P$, the statement $(\forall x \in \emptyset)(P(x))$. This can be argued as follows: Assume for contradiction $\neg [(\forall ...
24
votes
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
30 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
47 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
46 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
245 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
218 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
45 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
35 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
36 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
68 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 ...
25
votes
10answers
770 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
85 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
114 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
54 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
31 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
55 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
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3answers
116 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
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2answers
110 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
50 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
87 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 ...