For posts seeking explanation or clarification of a specific step in a proof. "Please explain this proof" is off topic (too broad, missing context). Instead, the question must identify precisely which step in the proof requires explanation, and why so. This should not be the only tag for a question, and should not be used to circumvent site policies regarding duplicate questions.
Questions tagged [proof-explanation]
11824 questions
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Proof $($by contradiction$)$
can anyone please explain through these? If so, I would really appreciate it. I think one, if not both, are proof by contradiction.
1) Suppose that m and n are negative integers with $m > n$.
Prove that $\sqrt{(m^2 + n^2)} \neq −(m + n)$.
2) Suppose…
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R is the relation on Q given by xRy iff x = $3^{k} y$, for some k ∈ Z
I Am studying my teacher's notes on our practice exam, And I am stuck on this part of the question where we are trying to prove that the relation is symmetric.
Symmetry: Let x, y ∈ Q. Suppose xRy. Then ${x = 3^{k}
y}$, for some k ∈ Z. Thus,
y =…
Jr194
- 109
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1 answer
Basel problem Euler's proof confusion.
This is from a book im reading.
$ 1-\frac{x^2}{3!}+ \frac{x^2}{5!} -\frac{x^2}{7!}+ \frac{x^2}{9!}-...$
$= [1-\frac{x^2}{\pi^2}]\ [1-\frac{x^2}{4\pi^2}]\ [1-\frac{x^2}{9\pi^2}]\ [1-\frac{x^2}{16\pi^2}]... $
$= 1-(\frac{1}{\pi^2}+ \frac{1}{4\pi^2}+…
Asim
- 328
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Confusing proof for $\sqrt{2}$ is irrational
Confusing $\sqrt{2}$ is irrational proof
Let $a, b$ be arbitrary positive integers.
Then $2b^{2}$ is divisible by 2 an odd number of times, while $a^{2}$ is
divisible by 2 an even number of
times, so $2b^{2} \neq a^{2}$ and therefore $|2b^{2}a^{2}|…
Andrew
- 385
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1 answer
Help with the Following Vacuous Proof
I'm working on the following proof:
Prove that if x, y, and z are three real numbers such that $x^2+y^2+z^20$.
I know that the proof is meant to be either vacuously or trivially true, and since $\exists x,y,z \in\Bbb R$ such…
user601846
- 111
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Using Sterling's Formula to prove log(n!) = Θ(nlogn)
I'm reviewing a proof here
and I'm having trouble understanding how to get to step 1. I know Sterlings formula is sqrt(2pi*n) * (n/e)^n
but where did they get (1 + o(1/n) from in step 1?
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Need help understanding this proof (algorithm complexity)
I don't understand how it's valid to present this counterexample. It doesn't satisfy $f(n) = O(g(n))$
since $f(n)$ is not $O(g(n))$. $f(n) = \omega (2^n)$ if $f(n)$ is $2n$ and $g(n)$ is $n$.
So how is it valid to simply "Let $f(n) = 2n$ and…
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Clarification on existential quantifier
The statement:
$\forall a,b \in \mathbb{R}, \exists c,d \in \mathbb{R}$, such that if $ab\geq cd$, then $a\geq c$ and $b\geq d$.
First, does this mean that $a, b, c, d$ all have to be different real numbers or can they all be the same?
Also, Does…
etkachuk
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$[a]_m \subseteq [a]_n$
I am completely lost. I have tried using an element $x$ in $[a]_m$ such that $x \equiv a \pmod{n}$, and I know this means $x=a+nk$ for some integer $k$, but I do not know how to show this is a subset of $[a]_n$ or how to show $n|m$ from this. Please…
iSuckAtMath
- 159
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4 answers
Show that $\frac{x-1}{y-1} \leq \frac{x}{y}$
i assume that this is very simple, but i cannot figure out a solution.
can somebody give a proof for
$$\frac{x-1}{y-1} \leq \frac{x}{y}$$
given that $$ x \leq y$$
and
$$ x \gt 1$$
thank you very much.
nveo
- 103
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1 answer
Boundedness theorem, why the limit of subsequence lies in the interval at which the original sequence is bounded?
Theorem: if $f$ is continuous on closed interval $I$ then it is also bounded. The proof is given by contradiction, assuming that for any $n$ it is possible to find $x_n$ such that it is greater than n $f(x_n)>n$. It is proven to be impossible by…
user
- 1,412
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2 answers
Prove that a cut is not produced by rational number
Notice:
I have asked a similar question on this topic before here, but it partook an alternative approach which didn't lead me anywhere, so now I am going with the classic one, suggested by the author; these, however, are two different questions, on…
Misha.P
- 221
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Elaboration of Thm. 1.1 in chapter V in Richard H. Crowell.
The theorem and its proof is given below:
But I do not understand why $(\rho i)_{*} = $ identity implies that $\rho_{*}$ is onto. Could anyone explain this for me please?
Intuition
- 3,269
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1 answer
Cover an open set of $\mathbb{R}^n.$
I have difficulty understanding the following result.
Theorem. Let $A\subseteq\mathbb{R}^n$ an open subset. Then for each $\delta>0$ the set $A$ is countable union of close cube having disjointed interiors and diagonal smaller than…
Jack J.
- 920
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What is the interpretation of $\frac{1+x_s}{1+x_u}$?
I tried to solve the problem
A ten year comparison between the United States and the Soviet Union
in terms of crop yields per acre revealed that when only planted
acreage is compared, Soviet yields were equal to 68 percent of United
States…
user685057