1.4 Intervals

Intervals: Special Subsets of Real Numbers

An interval is a set of real numbers between two endpoints.

Think: A section of the number line.

Intervals are subsets of $\mathbb{R}$ (real numbers).

Why they matter: Used extensively in calculus, inequalities, and real-world measurements.

Open Interval $(a, b)$

Definition: $(a, b) = \{x : a < x < b\}$

Key: Endpoints $a$ and $b$ are NOT included.

Use round brackets $()$

Think: Open doors - you can't touch the doors.

Half-Open Intervals

Type 1: $[a, b)$

Left endpoint included, right NOT included.

$[a, b) = \{x : a \leq x < b\}$

Type 2: $(a, b]$

Left endpoint NOT included, right included.

$(a, b] = \{x : a < x \leq b\}$

Length of an Interval

For any finite interval with endpoints $a$ and $b$ where $a < b$:

Length = $b - a$

Doesn't matter if endpoints are included or not!