Proof that rationals are uncountable

Let’s say I have a number q∈Q, i.e. ∀a∈Z.∀b∈N+.q=ab Now I look at a as an endless sequence (a0,a1,a2,…) where ai can be determined by the expression a=∑i=0ai⋅10i and I look at b as an endless sequence (b0,b1,b2,…), where bi can be determined by the expression b=∑i=0bi⋅10i Now I can combine these two sequences into … Read more

Where does this proof that 1=21 = 2 go wrong? (Expressing x2=∑xn=1xx^2=\sum_{n=1}^xx and differentiating) [duplicate]

This question already has answers here: Where is the flaw in this “proof” that 1=2? (Derivative of repeated addition) (10 answers) Closed 1 year ago. I saw a fake proof using the power rule to show that 1=2, thus disproving the power rule. It is obviously wrong but I can’t spot the error. It goes … Read more

proof that 11⋅2+12⋅3+⋯+1n(n−1)=32−1n \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \dots + \frac{1}{n(n-1)} = \frac{3}{2} – \frac{1}{n}

Proof that 11⋅2+12⋅3+⋯+1n(n−1)=32−1n by induction. Proof Base case: Statement clearly holds for n=1. Now assume that statement holds for some n=k and lets show that it implies n=k+1 holds. The proof: 11⋅2+12⋅3+⋯+1n(n−1)+1n(n+1)=32−1n+1n(n+1)=32−1n+1n−1n+1 =32−1n+1 Now the problem is I can’t find the error. The statement doesn’t clearly work for n=2. However, the assumption seems to be … Read more

Fake Proof that \mathrm{NP}^\mathrm{NP} = \mathrm{NP}\mathrm{NP}^\mathrm{NP} = \mathrm{NP}

I found this faulty proof of \newcommand{\NP}{\mathrm{NP}} \NP = \NP^{\NP}, where the tricky part is to proof that \NP ^{\NP} \subseteq \NP, and this is how it is realized: Take \NP^\NP machine that is a ND polytime TM1 running in O(P_1(n)) and assume the oracle as a ND polytime TM2 running in O(P_2(n)). Replace the … Read more

What’s wrong with this demonstration? (1 = -1) [duplicate]

This question already has answers here: Why $\sqrt{-1 \cdot {-1}} \neq \sqrt{-1}^2$? (14 answers) Closed 8 years ago. What’s wrong with this demonstration?: $$A \iff 1 = 1^1$$ $$A \implies 1 = 1^\frac{2}{2}$$ $$A \implies 1 = (1^2)^\frac{1}{2}$$ $$A \implies 1 = ((-1)^2)^\frac{1}{2}$$ $$A \implies 1 = (-1)^\frac{2}{2}$$ $$A \implies 1 = (-1)^1 = -1$$ … Read more

A false proof for the theorem about finite integral domains and fields

Related to another post of mine, right now I am looking at a proof for the theorem: A finite integral domain R is a field. In short, the proof uses a map fa:R→R defined for some a∈R by fa(x)=ax for all x∈R. Injectivity is easy to show and surjectivity is attained from injectivity and pigeonhole … Read more

What is the flaw in proving $i=0$ from $\ln(1)=0$?

Today, I was playing with numbers and I found this: $$\begin{align} \ln(1) = 0 \Rightarrow \ln((-1)^2) = 0 \Rightarrow 2\ln(-1) = 0 \Rightarrow 2\pi i = 0 \Rightarrow \boxed{i = 0} \end{align}$$ Apart from this I found another proof: $$\begin{align} \ln(1) = 0 \Rightarrow \ln(e^{2\pi i}) = 0 \Rightarrow 2\pi i = 0 \Rightarrow \boxed{i … Read more

Where is the error in derivative trick that shows 2=1? [duplicate]

This question already has answers here: Where is the flaw in this “proof” that 1=2? (Derivative of repeated addition) (10 answers) Closed 24 days ago. Today in class our professor showed us how “misinterpreting” what a derivative is can lead to bizarre results such as 2=1. He wrote the following on the board: n2=n+n+…+n (n times)2n=1+1+…+1 (n … Read more