Artificial intelligent assistant

infimum

answer Answers

ProphetesAI is thinking...

MindMap

Sources

1
infimum
infimum Math. (ɪnˈfaɪməm) [L., = lowest part, neut. of infimus lowest (see infima species).] The largest number that is less than or equal to each of a given set of real numbers; an analogous quantity for a subset of any other ordered set.1940 G. Birkhoff Lattice Theory ii. 16 We shall use the words... Oxford English Dictionary
prophetes.ai 0.0 3.0 0.0
2
Essential infimum and essential supremum - Wikipedia
Definition. As is often the case in measure-theoretic questions, the definition of essential supremum and infimum does not start by asking what a function does at points (that is, the image of ), but rather by asking for the set of points where equals a specific value (that is, the preimage of under ).. Let : be a real valued function defined on a set . The supremum of a function is ...
en.wikipedia.org 0.0 1.5 0.0
3
The infimum of the given set > What is the infimum of $A = \\{3^{2x} + 3^{\frac{1}{2x}} | x > 0\\}$? Without guessing, how do you determine the infimum of that set? Many advises tell us to guessing $x$, but for this...
By Arithmetic-Geometric inequality, \begin{align*} 3^{2x}+3^{1/(2x)}\geq 2(3^{2x}\cdot3^{1/(2x)})^{1/2}=2\cdot 3^{(2x+(2x)^{-1})/2}\geq 2\cdot 3^{((2x)(2x)^{-1})^{1/2}}=2\cdot 3=6, \end{align*} and the equality holds if and only if $3^{2x}=3^{1/(2x)}$ and $2x=(2x)^{-1}$. For $x>0$, we can take $x=1/...
prophetes.ai 0.0 0.90000004 0.0
4
Essential infimum and essential supremum
In mathematics, the concepts of essential infimum and essential supremum are related to the notions of infimum and supremum, but adapted to measure theory The essential infimum is defined in a similar way. wikipedia.org
en.wikipedia.org 0.0 0.6 0.0
5
Question based on infimum concept Somebody told me that infimum of smaller set is always bigger than infimum of bigger set .I couldnot understand it how ? How can it be justified ?
To see why, take the infimum $m$ of the bigger set $S$. For all $s\in S$, $m \leq s$ by definition. That is, $m$ is a lower bound on $T$ -- in particular, it is not greater than the infimum of $T$, which is the _greatest_ lower bound for $T$ possible.
prophetes.ai 0.0 0.6 0.0
6
Infimum of a certain set Let $A$ be the set $$A=\left\\{ |\sin(n)| : n\in\mathbb{N} \right\\},$$ then what is the infimum of $A$? I think the infimum is a positive quantity. I tried with the graph $$f(x)=|\sin(x)|,$$ ...
First, note that the zeroes of $\sin x$ are precisely the integer multiples of $\pi$. By Hurwitz' Theorem), there are infinitely many pairs of relatively prime integers $a, b$ such that $$\left\vert\pi - \frac{a}{b}\right\vert < \frac{1}{\sqrt{5} b^2},$$ and in particular we can approximate $\pi$ ar...
prophetes.ai 0.0 0.3 0.0
7
How do I find the infimum of a function using Calculus III techniques? > Let $f(x, y, z) = \frac{x + y + z}{2} - \sqrt{xyz}$. Find the infimum of the function where the domain is restricted to the first quadrant. The...
Take all the partials: ∂f/∂x = 1/2 - √(yz/x)/2 = 0 ∂f/∂y = 1/2 - √(xz/y)/2 = 0 ∂f/∂z = 1/2 - √(xy/z)/2 = 0 solving these yields the following: 1 = √(yz/x) thus yz=x 1 = √(zx/y) thus zx=y 1 = √(yx/z) thus xy=z multiplying all these equations yields: (xyz)^2 = (xyz) thus either xyz = 1 or xyz = 0 say ...
prophetes.ai 0.0 0.3 0.0
8
Can we use a limit argument to prove that an infimum = some value? I want to show that the infimum of the set containing the terms of the harmonic sequence is 0. Can I simply argue that because the harmonic sequence c...
Not exactly; the convergence alone does not ensure that conclusion. Indeed, there is a converging sequence $(x_{n})$ with limit not $\inf_{n}x_{n}$ (what is an example?). If $x_{n} := 1/n$, then $(x_{n})$ is decreasing ; so it just so happened that $\inf_{n}x_{n} = \lim_{n}x_{n}$ (Try to prove this;...
prophetes.ai 0.0 0.3 0.0
9
Supremum and infimum of a certain sequence What is the supremum and infimum of $$\\{(-1)^n-\dfrac1n\\}$$ And $n$ belongs to natural numbers .
The infimum (for odd $n$) is obtained for $n = 1$, that is, $1 - 1 = 0$. So, for the given sequence $\big\\{(-1)^n - \dfrac{1}{n}\big\\}$, the infimum is $-2$ and the supremum is $1$.
prophetes.ai 0.0 0.3 0.0
10
How can I proof the infimum and supremum of this set? $E = \\{{x+y : x,y \in\Bbb R_{>0}}$} I was able to figure out that this set does not have a supremum, but I am not able to prove it. Also, how can I prove the inf...
Assume that the $\sup E$ exists, thus there is some number $b \in R_{>0}$ s.t $\forall(x+y), x,y \in R_{>0}$ then $b \geq (x+y)$, but then $b+1$ is a sum of two numbers in $R_{>0}$ and $b+1>b$, contradiction.
prophetes.ai 0.0 0.3 0.0
11
To show infimum of set is 0 I have read Archimedean property. > The exercise has question to find the infimum of set $A = \\{1/n: n \in \mathbb N\\}$. Intuitively I see infimum to be $0$. But I need hint to prove t...
Let $\epsilon >0$ For the Archimedean property, there exist $N \in \mathbb{N}$, such that $N\epsilon>1 $, so $\frac{1}{N}< \epsilon$
prophetes.ai 0.0 0.0 0.0
12
Supremum and infimum of a set of reals Find the supremum and infimum of the following set $$ A:= \left\\{1 + \frac{(-1)^n}{n} \mid n\in \mathbb{N}\right\\}.$$ Here if $n$ is near infinity then the fraction part is $0$...
For a set $A=\\{a_n~:~n\in\mathbb N\\}$ the supremum and infimum is not the same as the limes superior or limes inferior of the sequence $(a_n)_n$.
prophetes.ai 0.0 0.0 0.0
13
Find the supremum and infimum of the set $S = \{ \sqrt {n^2 + 1} - n: n \in \mathbb{N}\}$ Find the supremum and infimum of the set $S = \\{ \sqrt {n^2 + 1} - n: n \in \mathbb{N} \\}.$ I know that the supremum is $\sqr...
The perennial classic $$ \sqrt{n^2+1}-n = \frac{1}{\sqrt{n^2+1}+n} $$ should convince you that you're correct: it's clear that this is a decreasing function of $n$, and smaller than $1/(2n)$, which also tends to zero.
prophetes.ai 0.0 0.0 0.0
14
What is the supremum and infimum of $f(x) = xe^{-x}$, where $x >0$? What is the supremum and infimum of $f(x) = x e^{-x}$, where $x > 0$? For the infimum, I do not know $\infty$. Zero equals what? How should I think ...
The infimum is zero because $f(x)$ is positive but comes arbitrarily close to zero for $x\to\infty$ and also for $x\to0$.
prophetes.ai 0.0 0.0 0.0
15
A set with a supremum and an infinum inside What is the name of a set $A$ that has its infimum $i \in A$ and a supremum $s \in A$? Examples: 1. $[0,1]$ has a supremum $1$ and an infimum $0$ which are inside $[0,1]...
Posets whose all non-empty subsets have an infimum and a supremum inside are exactly the finite totally ordered sets. Now, since all non-empty subsets have an infimum inside, $(D, \leq)$ is well ordered. In the same way, the dual of $(D, \leq)$ is well ordered.
prophetes.ai 0.0 0.0 0.0