Interval Notation

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)

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 $)

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) $