Interval & Set-Builder Notation

Q: Why does infinity get a parenthesis?

Infinity is not a number you can reach or include, so it is always excluded with a parenthesis: $[2, \infty)$, never $[2, \infty]$.

Q: What does the vertical bar mean in set-builder?

It reads as 'such that.' So $\{x \mid x > 0\}$ is 'the set of all x such that x is greater than 0.'

Q: Which bracket matches which inequality?

Include ($\leq$, $\geq$) matches a square bracket; strict ($ $) matches a parenthesis.

Interval and set-builder notation are two compact ways to write a set of numbers, like 'all x between -3 and 5.' They show up constantly in domain and range answers, so fluency in both pays off.

Interval notation uses brackets and parentheses; set-builder uses an inequality with a variable. The single rule that governs both: brackets include the endpoint, parentheses exclude it.

Interval notation and its brackets

A square bracket includes the endpoint; a parenthesis excludes it. So $[-3, 5)$ means from $-3$ (included) up to $5$ (not included).

Infinity always gets a parenthesis, never a bracket, because you can never reach it: $x \geq 2$ becomes $[2, \infty)$ .

Set-builder notation

Set-builder reads the set as a rule: $\{x \mid -3 < x \leq 5\}$ means 'the set of all $x$ such that $x$ is greater than $-3$ and at most $5$ .' The vertical bar means 'such that.'

The inequality signs carry the include/exclude information: $\leq$ and $\geq$ match brackets, while $<$ and $>$ match parentheses.

Worked examples

Example 1: inequality to interval

Write $-3 < x \leq 5$ in interval notation.

Left endpoint is strict

( \text{ at } -3

Right endpoint is included

] \text{ at } 5

Combine

(-3, 5]

Answer: $(-3, 5]$

Example 2: a one-sided set

Write $x \geq 2$ in interval notation.

Include 2 with a bracket

[2

Infinity always gets a parenthesis

\infty)

Combine

[2, \infty)

Answer: $[2, \infty)$

Example 3: interval to set-builder

Write $[-1, 4)$ in set-builder notation.

The bracket at -1 includes it

x \geq -1

The parenthesis at 4 excludes it

x < 4

Write both conditions as one rule

\{x \mid -1 \leq x < 4\}

Answer: $\{x \mid -1 \leq x < 4\}$

Try one yourself

Common questions

Why does infinity get a parenthesis?

Infinity is not a number you can reach or include, so it is always excluded with a parenthesis: $[2, \infty)$ , never $[2, \infty]$ .

What does the vertical bar mean in set-builder?

It reads as 'such that.' So $\{x \mid x > 0\}$ is 'the set of all x such that x is greater than 0.'

Which bracket matches which inequality?

Include ( $\leq$ , $\geq$ ) matches a square bracket; strict ( $<$ , $>$ ) matches a parenthesis.

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Interval & Set-Builder Notation

Interval notation and its brackets

Set-builder notation

Worked examples

Example 1: inequality to interval

Example 2: a one-sided set

Example 3: interval to set-builder

Try one yourself

Common questions

Keep learning

Domain & Range from Graphs and Equations

Key Features of Functions

Two-Step Inequalities