Mathematics Stack Exchange
Mathematics Stack Exchange

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.

An example of a fake proof is $$1=\sqrt{-1\cdot-1}=\sqrt{-1}\sqrt{-1}=i^2=-1$$ which fails because $\sqrt{xy}=\sqrt x\sqrt y$ does not hold if $x$ or $y$ is negative. Sometimes the proof may be presented as a puzzle, the challenge being to identify the flaw.

For asking about identifying flaws in general proofs ("spot the mistake"); the tag should instead be used.

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Lamport claims there is an error in Kelley's proof of the Schroeder-Bernstein theorem. What is it?

In section 4.1 of his note How to write a proof, Leslie Lamport mentions an error in Kelley's exposition of the Schroeder-Bernstein theorem: Some twenty years ago, I decided to write a proof of the Schroeder-Bernstein theorem for an introductory…
Marion
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"Too simple to be true"

As indicates the title, this question is about "proofs" of true statements which are short and/or look elegant but are wrong. I mean example like Cayley-Hamilton's theorem, which states that for a $n\times n$ matrix over $\Bbb C$, and $\chi$ its…
Davide Giraudo
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Is every parallelogram a rectangle ??

Let's say we have a parallelogram $\text{ABCD}$. $\triangle \text{ADC}$ and $\triangle \text{BCD}$ are on the same base and between two parallel lines $\text{AB}$ and $\text{CD}$, So, $$ar\triangle \text{ADC}=ar\triangle \text{BCD}$$ Now the things…
Harsh Kumar
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What is wrong with my "disproof" of Cantor's Theorem?

I cannot figure out what is wrong: We will attempt to show that $\mathcal{P} (\mathbb{N})$ is countable. We use the following corollary from Rudin's Principles of Mathematical Analysis, p. 29: Suppose $A$ is at most countable, and, for every…
Ovi
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Apparent inconsistency of Lebesgue measure

Studying the Lebesgue measure on the line I've found the following argument which concludes that $m(\mathbb{R}) < +\infty$ (where $m$ denotes the Lebesgue measure on $\mathbb{R}$). Obviously it must be flawed, but I haven't been able to find the…
Jonatan B. Bastos
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Can anything interesting be said about this fake proof?

The Facebook account called BestTheorems has posted the following. Can anything of interest be said about it that a casual reader might miss? Note that \begin{align} \small 2 & = \frac 2{3-2} = \cfrac 2 {3-\cfrac2 {3-2}} = \cfrac 2 {3 - \cfrac 2 {3…
Michael Hardy
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Why Are the Reals Uncountable?

Let us start by clarifying this a bit. I am aware of some proofs that irrationals/reals are uncountable. My issue comes by way of some properties of the reals. These issues can be summed up by the combination of the following questions: Is it…
akdom
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Tell me problems that can trick you

I am looking for problems that can easily lead the solver down the wrong path. For example take a circle and pick $N$, where $N>1$, points along its circumference and draw all the straight lines between them. No $3$ lines intersect at the same point…
Balazs Rau
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$1/i=i$. I must be wrong but why?

$$\frac{1}{i} = \frac{1}{\sqrt{-1}} = \frac{\sqrt{1}}{\sqrt{-1}} = \sqrt{\frac{1}{-1}} = \sqrt{-1} = i$$ I know this is wrong, but why? I often see people making simplifications such as $\frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}}$, and I would…
Tom
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False proof that $ρe^{iθ} = ρ$ and so complex numbers do not exist?

My professor showed the following false proof, which showed that complex numbers do not exist. We were told to find the point where an incorrect step was taken, but I could not find it. Here is the proof: (Complex numbers are of the form $\rho…
Madhav Nakar
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Trying to understand why circle area is not $2 \pi r^2$

I understand the reasoning behind $\pi r^2$ for a circle area however I'd like to know what is wrong with the reasoning below: The area of a square is like a line, the height (one dimension, length) placed several times next to each other up to the…
ByteFlinger
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Fake proofs using matrices

Having gone through the 16-page-list of questions tagged fake-proofs, and going though both the relevant MSE Question and Wikipedia page, I didn't find a single fake proof that involved matrices. So the question (or challange) here is: what are some…
sTertooy
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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 that $n^2 = n$. Dividing both sides by $n$…
Chris
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Historical Mistake of Assuming Measurability

I am recently reading about Fourier transforms and convolutions. It was a surprise to me that it takes quite several paragraphs to prove the measurability of innocent looking $f(x-y)$ (reference: proof that $\hat{f}(x,y)=f(x-y)$ is measurable if $f$…
温泽海
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Seems that I just proved $2=4$.

Solving $x^{x^{x^{.^{.^.}}}}=2\Rightarrow x^2=2\Rightarrow x=\sqrt 2$. Solving $x^{x^{x^{.^{.^.}}}}=4\Rightarrow x^4=4\Rightarrow x=\sqrt 2$. Therefore, $\sqrt 2^{\sqrt 2^{\sqrt 2^{.^{.^.}}}}=2$ and $\sqrt 2^{\sqrt 2^{\sqrt…
JSCB
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