# Questions tagged [fake-proofs]

Seemingly flawless arguments are often presented to prove obvious fallacies (such as 0=1). This is the appropriate tag to use when asking "Where is the proof wrong?" about proofs of such obvious fallacies.

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### Circumference of an ellipse is π(a + b)?

I know that a formula for an ellipse in terms of elementary functions doesn't exist. But I seem to have found an intuitive false-proof that the circumference of an ellipse with axes $a$ and $b$ equals ...
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### What is wrong with my argument that every group of order $pq$ is abelian?

Question: Prove that a group $G$ of order 143 is abelian. Solution: $G$ contains subgroups $H$ and $K$ of orders 11 and 13, respectively. The subgroups $G/H$ and $G/K$ have orders 13 and 11, ...
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### What's wrong with this integration over the volume of a sphere (from Gauss' Theorem)?

I'm struggling to find my mistake in the following problem: Let $S$ be a sphere of radius $R$ with center at the origin. Let $f(x,y,z) = 3z^2$. Find the integral of $f$ over the volume of the ...
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### Incorrect proof for $e$ idempotent $\Rightarrow$ $eA$ projective

I have seen several proofs for $e$ idempotent $\Rightarrow$ $eA$ projective where $A$ is an algebra. I tried something different and produced a proof without using the fact that $e$ is idempotent (so ...
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### Spot the error in the following "proof" by induction. [duplicate]

The following statement is false, but still, a proof is given. I need to determine the error. "In any set of $n≥1$ circles in $R^2$, all circles have the same radius". Base case: If $n=1$, ...
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### Prime avoidance for graded Noetherian ring with infinite residue field.

This question is related to this question. Let $R$ be a Noetherian local ring with infinite residue field. Let $\text{gr}_IR$ denote the associated graded ring $\bigoplus_{n=0}^{\infty} I^n/I^{n+1}$. ...
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### I can't find the mistake in my proof

I thought about something in power series and I proved a theorem that I have never seen before: The theorem: Let \begin{align} \sum_{n=1}^{\infty} a_nx^n \end{align} be a power series, with a ...
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### setting epsilon in sequence limit to the absolute value difference

Sequence is said to have a limit $L$ if: $$\forall\epsilon\in\mathbb{R}>0.\exists N\in\mathbb{R}.\forall n\in\mathbb{R}>N.\left| a_{n}-L \right|<\epsilon$$ since the absolute value is a ...
1 vote
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### How can I find the determinant of this $2\times 2$ block matrix directly?

I wish to explicitly compute the determinant of $$K = \begin{pmatrix} I & A \\ B & 0 \end{pmatrix}$$ where $A,B$ are $n\times n$ matrices and $I$ is the $n\times n$ identity, using the ...
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### Let $f :\mathbb R \rightarrow\mathbb R$ be continuous function . Then prove that $(0,1)$ cannot be the image of $(0,1]$?

Let $f :\mathbb R \rightarrow\mathbb R$ be continuous function. Then prove that $(0,1)$ cannot be the image of $(0,1]$ i.e $f((0,1])\neq (0,1)$ ? My solution: Let us assume the contrary that, ...
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### False proof that $\tfrac{0}{0} = 2.5$ [closed]

I was just playing around with the quadratic formula and wondered what if the a coefficient is $0$, so I took a normal equation like $2x-5=0$ and slapped a $0x^2$ term in front of it to get a ...
1 vote
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### Not getting where is the mistake in the solution of a problem in Linear Algebra.

The set of all $n×n$ matrices having trace equal to zero is a subspace $W$ of $M_{n×n}(F).$ Find a basis for $W.$ What is the dimension of $W?$ To solve this problem, I first noted that the standard ...
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### What's wrong with the following limit evaluation?

\begin{align*} \lim_{x \to \infty} \ln{x}&=\lim_{x \to \infty}\frac{\ln{x}}{x}\cdot x\\ &=\lim_{x \to \infty}\frac{\ln{x}}{x}\cdot \lim_{x \to \infty} x\\ &=\lim_{x \to \infty}\frac{(\ln{x}...
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### Show Tree Proof is Incorrect

It seems the part that says every vertex in H has degree at least two is not necessarily true. The paths could converge prior to seeing vertex v, thus resulting in at least one vertex having degree 1. ...
1 vote
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### Is this proof of Darboux's Theorem valid?

Let $f:\Bbb [a,b]\to \Bbb R$ satisfy the following : (i) $f(x)$ is derivable in $[a,b]$ (ii) $f'(a)=\alpha\neq f'(b)=\beta$ and (iii) $\exists \gamma\in (\alpha,\beta)$ then, there exists at least one ...
62 views