Interval Notation

Interval Notation

Introduction

Interval notation is a way to represent sets of real numbers.
It uses parentheses and brackets to indicate whether the endpoints are included or excluded.
Let $a$ and $b$ be two real numbers such that $a < b$ .

Open Interval

Notation: $(a, b)$
Definition: The set of real numbers between $a$ and $b$ , not including $a$ and $b$ .
Set builder notation: $\lbrace x \mid a < x < b \rbrace$
Graphical representation:
- Number line with points $a$ and $b$ .
- Open parentheses at $a$ and $b$ .
- Shaded region between $a$ and $b$ .
Parentheses indicate that the endpoints are excluded from the interval.

Closed Interval

Notation: $[a, b]$
Definition: The set of real numbers between $a$ and $b$ , including $a$ and $b$ .
Set builder notation: $\lbrace x \mid a \leq x \leq b \rbrace$
Graphical representation:
- Number line with points $a$ and $b$ .
- Square brackets at $a$ and $b$ .
- Shaded region between $a$ and $b$ .
Square brackets indicate that the endpoints are included in the interval.

Infinite Interval (Right)

Notation: $(a, \infty)$
Definition: The set of real numbers that are greater than $a$ .
Set builder notation: $\lbrace x \mid x > a \rbrace$
Graphical representation:
- Number line with point $a$ .
- Open parenthesis at $a$ .
- Shaded region to the right of $a$ .
The infinity symbol $\infty$ does not represent a real number; it indicates that the interval extends indefinitely to the right.

Infinite Interval (Left)

Notation: $(-\infty, b]$
Definition: The set of real numbers that are less than or equal to $b$ .
Set builder notation: $\lbrace x \mid x \leq b \rbrace$
Graphical representation:
- Number line with point $b$ .
- Square bracket at $b$ .
- Shaded region to the left of $b$ .
The negative infinity symbol $\infty$ indicates that the interval extends indefinitely to the left.

Parentheses vs. Brackets

Parentheses: Indicate endpoints that are not included in the interval.
Square brackets: Indicate endpoints that are included in the interval.
Parentheses are always used with positive infinity $(+\infty)$ or negative infinity $(-\infty)$ .

Nine Possible Types of Intervals

Let $a$ and $b$ be real numbers such that $a < b$ .

Interval Notation	Set Builder Notation	Graph
$(a, b)$	$\lbrace x \mid a < x < b \rbrace$	Number line with open parentheses at $a$ and $b$ , shaded between.
$[a, b]$	$\lbrace x \mid a \leq x \leq b \rbrace$	Number line with closed brackets at $a$ and $b$ , shaded between.
$[a, b)$	$\lbrace x \mid a \leq x < b \rbrace$	Number line with a closed bracket at $a$ and an open parenthesis at $b$ , shaded between.
$(a, b]$	$\lbrace x \mid a < x \leq b \rbrace$	Number line with an open parenthesis at $a$ and a closed bracket at $b$ , shaded between.
$(a, \infty)$	$\lbrace x \mid x > a \rbrace$	Number line with an open parenthesis at $a$ , shaded to the right.
$[a, \infty)$	$\lbrace x \mid x \geq a \rbrace$	Number line with a closed bracket at $a$ , shaded to the right.
$(-\infty, b)$	$\lbrace x \mid x < b \rbrace$	Number line with an open parenthesis at $b$ , shaded to the left.
$(-\infty, b]$	$\lbrace x \mid x \leq b \rbrace$	Number line with a closed bracket at $b$ , shaded to the left.
$(-\infty, \infty)$	$\lbrace x \mid x \in \mathbb{R} \rbrace$	Fully shaded number line.

Examples

Part a: $[-2, 5)$

Set builder notation: $\lbrace x \mid -2 \leq x < 5 \rbrace$
Graphical representation:
- Bracket at $-2$ (inclusive).
- Parenthesis at $5$ (exclusive).
- Shade between $-2$ and $5$ .

Part b: $[1, 3.5]$

Set builder notation: $\lbrace x \mid 1 \leq x \leq 3.5 \rbrace$
Graphical representation:
- Bracket at $1$ (inclusive).
- Bracket at $3.5$ (inclusive).
- Shade between $1$ and $3.5$ .

Part c: $(-\infty, -1)$

Set builder notation: $\lbrace x \mid x < -1 \rbrace$
Graphical representation:
- Parenthesis at $-1$ (exclusive).
- Shade to the left of $-1$ with an arrow to indicate that it continues to negative infinity.