Questions tagged [interval-arithmetic]

Interval arithmetic is the arithmetic of quantities that lie within specified ranges (i.e., intervals) instead of having definite known values.

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Question about an absolute value inequality.

The question states: Given $g(x) = |x-2| - |x| +2$, express $g(x)$ without absolute value bars if $x$ is in given interval - $1.$ $[2, +\infty)$ $2.$ $[ -\infty, 0)$ $3.$ $[0,2)$ So I found the ...
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Interval and percentage algorithm

I have a problem that I can not solve. I want to create the following algorithm: For an interval [a, b]. If a variable x (between a and b) is close to the center of the interval we will output 100% ...
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28 views

How to write a subset with index

I have a difficulty to define the start and end index of a subset. I have $x_i$, $i=1, ..., |\phi_M|$ an element of the main set $\phi_M$, Now I have to define subsets of $\phi_M$ but I have doubt on ...
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Will linear interpolation of a random value from one range to another 'preserve' randomness?

I have generated a number z from a range (x,y) and want to map z to another range (p,q) This can be achieved using linear interpolation as seen here: https://www.mathworks.com/matlabcentral/answers/...
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How are positive intervals calculated?

Give these equations $N=p^4-p^3+161*p$ $N+(n/2)^2=M^2$ $p*(p+n)=161$ How to calculate the intervals in which $n>0$ and $p>0$ and $M>0$ and $N>0$ ? $n$ and $N$ and $p$ and $M$ not ...
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can some please correct the following table and explain my questions? [closed]

as the title says, enter image description here I'm having trouble understanding the type and description of some intervals in this table
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Why interval property fails for this types of problems: $|y-a|>b, a,b\in\mathbb{R}?$

The interval property reads: Given $b\in \mathbb{R}$ with $b>0$, we have $|x|<b$ if and only if $-b\lt x \lt b$. Corollary reads: Let $a,b\in \mathbb{R}$ with $b\gt 0$ be given. For all $y\in \...
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Proof of Theorem 1.1.4 in Durrett — measure on the real line

This question arose from the proof of theorem 1.1.4 in Durrett 5th Edition. Firstly, measure on $(\mathbb{R},\mathcal{R})$are defined by giving a Stieltjes measure function with the following ...
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1answer
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Why do second or higher derivatives work for finding concavity and inflection points?

Say we have the function $f(x)=(x-2)^3+3$, whose graph is and we want to find at what regions does $f$ have a positive/negative concavity, and where the inflection points are. I learned to answer ...
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Meaning of a bracket facing not the interval

What does it mean by $]a,b]$ ? And what is the notation called? First time to for me to see this expression and I cannot get the meaning
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Dividing the continuum and adding recurring decimals

Let us separate all of the positive numbers contained within the integer "1" into two congruent sets: [0,0.5), [0.5,1) In this case, the value "1" is excluded from the sets, we can consider this ...
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Network flow and uncertainty in supply/demand

I am trying to learn how to study uncertainty in supply in a network flow problem. Specifically, I am using network simplex method on an undirected graph with a bunch of nodes with source/supply and a ...
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Range of functions

What is the range of the function: $$f(x,y)=\frac{7}{x^2+y^2+1}$$ I was thinking maybe, given that $x^2+y^2$ does not equal $-1$, then it could be any number from 0 to infinity? Or is that wrong?
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compare two intervals

I want to compare between two interval. What is the formula that I can use to compare two intervals and returns the biggest one? for example: interval1= [89.90, 92.25] interval2= [89.30, 93] ...
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1answer
46 views

Existence condition of solution for a system of nonlinear equations

I have the following system of nonlinear equations: $$\left\{ \begin{align} & \dfrac{-a}{w^2}\sin(wT) + \dfrac{T}{w}[a\cos(wT) - b\sin(wT)] + c = 0 \\ & \dfrac{b}{w}\sin(wT) + T[a\sin(wT) + b\...
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1answer
90 views

Sigma notation with indefinite limits

I would like to sum over a partial interval: $$ \sum_{t = -k}^{0}\beta_{k}, $$ where $\beta$ could be any constant (e.g., $\beta = 5$). I want to express an additive sum from $-k$ to $0$. The ...
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Why does $g(x)=(x^2−1)/3$ map $[-1,1]$ to $[-1/3, 0]$?

This is a particular part of a fixed-point problem I don't understand. The problem says: Let $g(x)=\dfrac{(x^2−1)}3$. Does $g$ have a fixed point on the interval $[−1,1]$? So naturally we have to ...
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57 views

Sigma notation with a variable in the lower limit

I have witnessed the following mathematical notation in various scholarly journals: $$ \sum_{t}^{} $$ where the lower limit only contains the variable $t$. In these circumstances, the person is ...
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How do you determine intervals of decrease and increase nonlaboriously?

I know points of increase and decrease can be found using the derivative. To take an example of $f(x) = x^2$, we have $\frac{d}{dx}f(x) = 2x$. The slope is constant. When $\frac{d}{dx}f(x) = 0$, we ...
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Is it okay to write $[0,\infty]$?

In my tuition school, a teacher wrote $[0, \infty]$ and i asked him if it is ever possible to write $\infty]$. He sternly replied not to question about silly things and focus on what is being taught. ...
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Give intervals for the following

$\{ X \}$ is fractional part of $X$. $[ X ]$ is greatest integer function (integral part of $X$) $| X |$ is modulus of $X$ 1) $X^2 > \{ X \}$ 2) $X^2 > [ X ]$ 3) $X^2 ≥ 4$ 4) $X^2 > ...
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Interval arithmetic problems

I'm trying to solve a few examples with intervals. The problem is I'm confused, and I couldn't find much info on the internet that could help me. I'll list the exercises and share my thoughts and ...
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Looking for a proper math notation to deal with intervals

Say we have two sets like this: $A = \{a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8,a_9,a_{10}\}$ and $B = \{b_1,b_2,b_3,b_4,b_5,b_6,b_7,b_8,b_9,b_{10}\}$ I want to be able to build something like this: $[(a_{1},...
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Can we find the domain of a function when we are given its range and its notation?

Let's say we have a function $f(x)=x^2$, we obviously know its domain is all real numbers (from the notation), but what if we restrict the outputs to a chosen interval, lets say $(2, +\infty)$, which ...
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Symbol for intervals

Set of integers is denoted by the symbol $\mathbb Z$, $\mathbb Q[x]$ stands for univariate polynomials over rationals, etc. Is there a symbol which indicates the set of all open (or closed) intervals ...
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1answer
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The set of all possible values of $AC$ is an open interval $(m,n)$

Side $\overline{AB}$ of $\triangle ABC$ has length $10$. The bisector of angle $A$ meets $\overline{BC}$ at $D$, and $CD = 3$. The set of all possible values of $AC$ is an open interval $(m,n)$. What ...
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Silly question on range calculation

A few days ago when I was driving, I noticed my car's odometer was about to hit 99,900 miles (my odometer ONLY shows an integer, no decimal points; while my trip meter always gives 1 decimal point). ...
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Why do we take only intervals for solution of differential equation [closed]

Suppose the solution to my equation blows up at 0 and is well defined everywhere else. Then why can't we just take R-{0} as the domain of the function? Also can someone give an example of a nonzero ...
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Minimum Sample size between two samples for a specified confidence interval/level with sigma known

Hi math stack exchange, I came across the following question and found it quite interesting and am struggling to solve it. I haven't seen anything like it because it is both two sample and has sigma ...
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$I$ is a union of intervals such that there do no exist 2 points in $I$ with difference $1/12$.Prove the sum of lengths of intervals is at most $1/2$

Consider $I$ a union of disjoint intervals inlcuded in the interval $[0, 1]$ such that there do no exist 2 points in $I$ situated at distance $1/12$. Prove that the sum of the lengths of the intervals ...
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659 views

Proving the interval $(0, 1)$ and $(1, 3)$ have the same cardinality.

Prop: Show that the intervals $(0, 1)$ and $(1, 3)$ have the same cardinality. What I have tried: I showed that the intervals $[2, 4]$ and $[0, 5]$ have equal cardinality by creating a function $F(x) ...
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Constructing an upper confidence limit for $σ^2$

Suppose that $X_1, ..., X_n$ form a random sample from the normal distribution with unknown mean µ and unknown standard deviation σ. Construct an upper confidence limit U(X1, ..., Xn) for $σ^2$ such ...
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How can I to prove the equality of intervals of open intervals with the equality of the closed interval

How can I to prove the equality between the following intersection of of open intervals with the following closed interval? $$ \bigcap_{n=1}^{\infty} \left(\frac{-1}n , 1+\frac{1}n\right) = [0,1] $$...
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If $k\in\mathbb{N}^*$, Give the interval of length $\frac{1}{10^k}$ where all rationals have zero at k-th decimal

I need help with this. The teacher defined $\mathbb{N}^*$ as $\mathbb{N}^* =\{1,2,...\}$. I tried to express an interval in decimal expansion like: $$a<r<a+\frac{1}{10^k}$$ Where $a\in \...
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meaning of multi-interval

According to the following definition, a multi-interval is just a line segment for n=1, a square region for n=2, a cubic for n=3, and so on so forth, correct?
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How many prime numbers in a given interval?

Is there any algorithm or a technique to calculate how many prime numbers lie in a given closed interval [a1, an], knowing the values of a1 and an, with a1,an ∈ ℕ? Example: [2, 10] --> 4 prime ...
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Verification of proof on Bounded Variation.

Let $f:[-1,1]\rightarrow \mathbb R$ be a function given by $f(x) = \begin{cases} x^2\cos(\frac{1}{x}), & \text{if $x\neq 0$ } \\ 0, & \text{if $x=0$ } \end{cases}$then $(a)$ $f$ is of ...
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Intervals and polynomials [closed]

If a polynomial is evaluated over an interval (by the use of interval arithmetic and Horner's method), what does the result say about the polynomial's roots? I mean, if the resulting interval ...
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1answer
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What is the difference between discrete interval and continous interval

I'm looking at my math textbook and it says for discrete distribution where the range is from a to b, $$f(x) = \frac{1}{b-a+1}$$ While for continuous distribution it states that $$f(x) = \frac{1}{b-a}...
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choosing error bounds for factors of a product so the product falls within a given error bound

Given $a,b,n\in\mathbb{R}$ such that $n=ab$, suppose I want to find a numerical approximation of $n$ in the form of an interval that the exact value of $n$ must fall in, that I want the length of the ...
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Intuition for interval subtraction

I'm working on a problem that involves some interval arithmetic, and while I can wrap my head around the reasoning for interval addition: the minimum value the sum could be is the sum of the two ...
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evaluation of the sum $\sum_{a=1}^{p-1} \left\lfloor \frac{\left\lfloor{v/p}\right\rfloor-a}{q}\right\rfloor$

Looking for the evaluation of the sum $$\sum_{a=1}^{p-1} \left\lfloor \frac{\left\lfloor{v/p}\right\rfloor-a}{q}\right\rfloor$$ where $p < q$, $p$ and $q$ are primes, and $v = (N \mod{p*q})$ where ...
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Given an interval around each rational number: How to find a real number which is in none of these intervals?

There are countable infinite many rational numbers, so it is possible to "count" them: There are many ways to do this; just pick one. If an order is chosen, there is an $n$th rational number. Each of ...
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Someone can explain this interval?

I know that $ | a | < b$ it is a interval, where $ a $ and $ b $ is an generic expression, but how i can understand this? I want to transform this on the interval like this $ a < b < -a $ (or ...
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Constructing a closed interval from open intervals of real numbers?

Is it possible to construct a closed interval, say [0,1] using only open intervals? Ah, sorry. I should have said. Using the three set operations (union, intersection, complement) and open intervals ...
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Is there a term for the interval [-1.0, 1.0]?

When representing data, it is sometimes appropriate to normalize (typically rational) numbers to the range $$[0.0, 1.0]$$ For variables that have symmetric extremes (min, max) around a meaningful zero,...
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$I_1, I_2, I_3$ intervals of even length, such that intersection is odd length

Does there exists three intervals $I_1,I_2,I_3$ each of length as an even integer such that $I_1 \cap I_2$ , $I_2 \cap I_3$ and $I_3 \cap I_1$ are of length as an odd integer? Tried few examples and ...
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Notation Convention for integer in a certain range

I am wondering what notation I should use when writing that some variable is an integer within some range. What is the most common way to do this? Here are some ideas I have but I'm not sure what the ...
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557 views

Let $x \in \Bbb{R}$. Prove $1 \leq x \leq 2$ if and only if $1 \leq x \leq 1 + \frac{1}{n}$ for some $n \in \mathbb{N}$?

The title is self-explanatory. How would I go about proving "$1 \leq x \leq 2$ if and only if $1 \leq x \leq 1 + \frac{1}{n}$ for some $n \in \mathbb{N}$"?
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Show that [4,5] and [4,12] are equivalent or have the same cardinality [closed]

I have 2 intervals, $[4,5]$ and $[4,12]$, so far I've just have found the bijection between the two which is $f: [4,5] \to [4,12]$, $f(x) = 8x-28$. If you can, make a full example proof step by step. ...