2.2 Interval notation

A set of real numbers between two endpoints is written as an interval. The only thing to keep track of is whether the endpoints themselves are included.


Interval	Meaning	In set-builder notation

\([a,b]\)	closed: both ends included	\(\{x\in \mathbb {R} : a\leq x\leq b\}\)
\((a,b)\)	open: neither end included	\(\{x\in \mathbb {R} : a< x< b\}\)
\([a,b)\)	half-open	\(\{x\in \mathbb {R} : a\leq x< b\}\)
\((a,b]\)	half-open	\(\{x\in \mathbb {R} : a< x\leq b\}\)
\([a,\infty )\)	unbounded above	\(\{x\in \mathbb {R} : x\geq a\}\)
\((-\infty ,b)\)	unbounded below	\(\{x\in \mathbb {R} : x< b\}\)

Table 3: Interval notation. A square bracket includes the endpoint, a round one excludes it.

Note 2.7. \(\infty \) is not a number and can never be included, so it always takes a round bracket: \([2,\infty )\), never \([2,\infty ]\).

Example 2.8. Write each of the following in interval notation.

(a).: All real numbers from \(-1\) up to and including \(4\).
(b).: \(\{x\in \mathbb {R} : x>7\}\)
(c).: All real numbers at most \(0\) or at least \(3\).

Solution. (a) \(-1\) is excluded by “from”, \(4\) is included by “and including”: \([-1,4]\) if \(-1\) is meant to be included, and here “from \(-1\)” does include it, so \[[-1,4].\]

(b) Strictly greater, so the endpoint is out and there is no upper limit: \[(7,\infty ).\]

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