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Interval Notation

Tool to convert to write intervals into inequalities and vice versa. Intervals/Ranges notation represent sets of numbers between two values.

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Interval Notation -

Tag(s) : Mathematics

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# Interval Notation

## Inequality to Interval Converter

 Locale Notation European ]…;…[ American (…;…)

## Set Notation Converter

 Convert to Inequality … < … Interval notation […;…] (only one)
 Locale Notation European ]…;…[ American (…;…)

### What is an interval? (Definition

An interval is a notation which makes it possible to define a set of real numbers included between a lower limit (minimum admissible value) and an upper limit (maximum admissible value).

There are 3 types of intervals (taking ${a, b} \in \mathbb {R}$ with $a < b$) and a variable $x \in \mathbb {R}$:

— open interval (or unclosed interval)

Example: $a < x < b = ] a; b [$

— closed interval (or unopened interval)

Example: $a \leq x \leq b = [a, b]$

— half-open (or half-closed) interval

Example: $a < x \leq b = ] a, b ]$ (half-open interval on the left or half-closed interval on the right)

Example: $a \leq x < b = [ a, b [$ (half-closed interval on the left or half-open interval on the right)

### How to convert an interval into an inequation?

Here is the list of different types of intervals and the corresponding inequalities:

$] a ; b [ \iff a < x < b \quad$ (or $b > x > a$)

$[ a ; b [ \iff a \leq x < b \quad$ (or $b > x \geq a$)

$] a ; b ] \iff a < x \leq b \quad$ (or $b \geq x > a$)

$[ a ; b ] \iff a \leq x \leq b \quad$ (or $b \geq x \geq a$)

A shorter writing of the inequality is possible if $a$ or $b$ has for value $\infty$

$] -\infty ; b [ \iff x < b \quad$ (or $b > x$)

$] -\infty ; b ] \iff x \leq b \quad$ (or $b \geq x$)

$] a ; +\infty [ \iff x > a \quad$ (or $a < x$)

$[ a ; +\infty [ \iff x \geq a \quad$ (or $a \leq x$)

### What is the European notation and the American notation?

There are two schools, two ways of writing which differ in the notation of open intervals.

European notation, which uses square brackets everywhere and denotes open intervals with an outward bracket: $[a; b [$

American notation, which uses either square brackets for closed intervals and denotes open intervals with a parenthesis: $[a; b)$

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Interval Notation on dCode.fr [online website], retrieved on 2023-09-21, https://www.dcode.fr/interval-notation

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