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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Why is this $0 = 1$ proof wrong?

$0 = 0 + 0 + 0 + ...$ $0 = (1 - 1) + (1 - 1) + (1 - 1) + ...$ $0 = 1 - 1 + 1 - 1 + 1 - 1 + ...$ $0 = 1 + (-1 + 1) + (-1 + 1) + (-1 + ...$ $0 = 1 + 0 + 0 + 0 + ...$ $0 = 1$ I can't really tell what is obviously wrong with this. It seems…
AureliusPhi
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Proof that $i=1=-1$( Mathematical Fallacy)

I thought of a proof that:- $$i=-1=1$$ Here is the proof :- $$ (i)^4=(i^2)^2 \\ = (\sqrt{-1})^4=(\sqrt{-1}^2)^2 \\ = (\sqrt{-1})^4=(-1)^2 \\ = (\sqrt{-1})^4=1 \\ \implies i^4=1 \\ \text{also} \\ -1^4=1 \\ \text{and} \\ 1^4=1 \\ \implies…
Mohd Saad
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Questions about the proof: "all positive integers are equal"

Others have expressed confusion about this proof on Stack and I have looked through every one of them because I don't want to post a duplicate post, however, none of them answer the questions I have and I am still confused. So we have to find what…
computerscienceisapain
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Is Hausdorff maximal principle and Kuratowski lemma same things?

In the John's Kelley "General Topology" (pages 32-34) there is definition of both of them and its look like its just different ways of notation of one proposition. But also there is something strange on page 34 that is called "proof" and perhaps is…
Andrey Komisarov
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Factoring cos, and other trig functions

EDIT: I misattributed the solution to factoring. The teacher in fact used a the trig identity: $\cos(A+B) = \cos(A)\cos(B) - \sin(A)\sin(B)$ I apologise for the time wasted on my expediant attempt at factoring trig. I am now more educated on trig…
Thomas J.
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Transformation Existence Proof: A Call for Critique

QUESTION Prove that there exists a $T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$ ATTEMPTED ANSWER Let $V$ and $W$ be finite-dimensional vector spaces over $F$. Let $A=\{a_1,\dots,a_l\}$ be a basis for $V'\subset V$, let…
Trancot
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What is wrong with this proof of $i = 0$?

This is a proof I made a year ago and at that time, I didn't see any problems with it. Could anyone point out what is wrong here? Consider the following expresion: $(-1)^{(4n+3)/2}$, where $n \in \Bbb Z_+$. We have that: …
Lê Thành Đạt
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What went wrong in proving $i=1$

I started with $$x=(-16)^{\frac{1}{2}}$$ $$x=(-16)^{\frac{2}{4}}$$ Since $$(a^m)^n=a^{mn}$$ we have: $$x=((-16)^2)^{\frac{1}{4}}$$ $$x=((16^2)^{\frac{1}{4}}$$ $$x=\sqrt{16}=4$$ Hence $$(-16)^{\frac{1}{2}}=4i=4$$ $$i=1$$
Ekaveera Gouribhatla
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What is the fallacy of this proof that $a=b$?

Let,you have an equation=$a^2-2ab+b^2$ This can be written in two ways- $$a^2-2ab+b^2\space \space \space \space \space \space \space \space \space \space \space \space \space \space \space b^2-2ab+a^2$$ And so, $$(a-b)^2=(b-a)^2$$ And so…
Soham
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Is this mathematical statement?

$\{\text{integers $n$ such that $n$ is even}\}$ It can be true/false so does that mean it's proposition/mathematical statement?
user3461957
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$1+1=0$ What am I doing wrong???!

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! $$1+1=$$ $$1+\sqrt{1}=$$ $$1+\sqrt{-1*-1}=$$ $$1+\sqrt{-1}*\sqrt{-1}=$$ $$1+i*i=$$ $$1-1=0$$
Jan Eerland
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Why does 2 equal 1?

A friend showed me this proof: Proof: 2 = 1 $$Let \space x= y$$ Multiply both sides by x: $$x^2= xy$$ Subtract $y^2$ from both sides: $$x^2-y^2= xy-y^2$$ Factor: $$(x+y)(x-y) = y(x-y)$$ Cancel out $(x-y)$ from both sides: $$(x+y) = y$$ Simplify…
user14069
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What, if anything, is wrong with this condensed proof in Daniel Solow

What, if anything, is wrong with the following condensed proof? enter image description here
Josue Furiano
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What is going wrong in this "proof" of $0=1$?

\begin{align} -20 &= -20\\ 16-36 &= 25-45\\ 4^2-4\times 9&=5^2-5\times 9\\ 4^2-4\times 9+81/4&=5^2-5\times 9+81/4\\ 4^2-4\times 9+(9/2)^2&=5^2-5\times 9+(9/2)^2\\ \end{align} Considering the formula $a^2+2ab+b^2=(a-b)^2$, one…
Animesh Sahu
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Is there any value for not defined?

\begin{align*} \sec x\cdot\cos5x+1&=0, \qquad 0
Fawad
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