Questions tagged [proof-explanation]

This tag is for readers who ask for explanation and clarification of some steps of a particular proof.

10133 questions
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Division by $0$ and its restrictions

Consider the following expression: $$\frac{1}{2} \div \frac{4}{x}$$ Over here, one would state the restriction as $x \neq 0 $, as that would result in division by $0$. But if we rearrange the expression, then: $$\begin{align} \frac12\div\frac4x &=…
8 answers

If $S$ is an infinite $\sigma$ algebra on $X$ then $S$ is not countable

I am going over a tutorial in my real analysis course. There is an proof in which I don't understand some parts of it. The proof relates to the following proposition: ($S$ - infinite $\sigma$-algebra on $X$) $\implies $ $S$ is…
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Compute polynomial $p(x)$ if $x^5=1,\, x\neq 1$ [reducing mod $\textit{simpler}$ multiples]

The following question was asked on a high school test, where the students were given a few minutes per question, at most: Given that, $$P(x)=x^{104}+x^{93}+x^{82}+x^{71}+1$$ and, $$Q(x)=x^4+x^3+x^2+x+1$$ what is the remainder of $P(x)$…
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What is the explanation for this visual proof of the sum of squares?

Supposedly the following proves the sum of the first-$n$-squares formula given the sum of the first $n$ numbers formula, but I don't understand it.
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Does Monty Hall logic apply to this real world situation?

I recently posted a tweet claiming I had encountered a real life Monty Hall dilemma. Based on the resulting discussion, I'm not sure I have. The Scenario I have 3 tacos (A,B,C) where tacos A and C are filled with beans, and taco B is filled with…
Will Cole
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In a proof by contradiction, how do we know the assumption is the cause of the contradiction?

In a proof by contradiction, how do we know the assumption is the cause of the contradiction? And not just the result of some other property more fundamental to numbers? In other words, how can we be sure we arrived at the contradiction because…
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Still struggling to understand vacuous truths

I know, I know, there are tons of questions on this -- I've read them all, it feels like. I don't understand why $(F \implies F) \equiv T$ and $(F \implies T) \equiv T$. One of the best examples I saw was showing how if you start out with a false…
10 answers

Why does this way of solving inequalities work?

Here is what I had to prove. Question: For positive reals $a$ and $b$ prove that $a^2+b^2 \geq 2ab$. Here is how my teacher did it: First assume that it is in fact, true that $a^2+b^2 \geq 2ab$. Therefore $a^2+b^2-2ab \geq 0$ . We have $(a-b)^2$…
The Cryptic Cat
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Is this a valid proof that there are infinitely many natural numbers?

I remember reading a simple proof that natural numbers are infinite which goes like the following: Let $ℕ$ be the set of natural numbers. Assume that $ℕ$ is finite. Now consider an arbitrary number $K$, where $K$ is the largest number in…
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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 $n\ge 0$. Base for $n=0$, $3n=3(0)=0$ Assume Induction Hypothesis: $3k…
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A subgroup of a cyclic group is cyclic - Understanding Proof

I'm having some trouble understanding the proof of the following theorem A subgroup of a cyclic group is cyclic I will list each step of the proof in my textbook and indicate the places that I'm confused and hopefully somewhere out there can clear…
8 answers

Reasoning that $ \sin2x=2 \sin x \cos x$

In mathcounts teacher told us to use the formula $ \sin2x=2 \sin x \cos x$. What's the math behind this formula that made it true? Can someone explain?
Commander Shepard
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Right adjoints preserve limits

In Awodey's book I read a slick proof that right adjoints preserve limits. If $F:\mathcal{C}\to \mathcal{D}$ and $G:\mathcal{D}\to \mathcal{C}$ is a pair of functors such that $(F,G)$ is an adjunction, then if $D:I\to \mathcal{D}$ is a diagram that…
4 answers

A function with a non-zero derivative, with an inverse function that has no derivative.

While studying calculus, I encountered the following statement: "Given a function $f(x)$ with $f'(x_0)\neq 0$, such that $f$ has an inverse in some neighborhood of $x_0$, and such that $f$ is continuous on said neighborhood, then $f^{-1}$ has a…
12 answers

How to prove that $\sqrt{2+\sqrt3}-\sqrt{2-\sqrt3}=\sqrt2$ without squaring both sides

I have been asked to prove: $$\sqrt{2+\sqrt3}-\sqrt{2-\sqrt3}=\sqrt2$$ Which I can easily do by converting the LHS to index form, then squaring it and simplifying it down to get 2, which is equal to the RHS squared, hence proved. However I know…
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