Writing down a set of numbers

Inequalities, interval notation and set-builder notation, and how to move between them.

A domain is a set of numbers, and sets of numbers turn up constantly once you start answering questions about functions. Saying the answer out loud is easy enough. Writing it in the form an examiner expects is a separate skill, and it is worth half an hour now to save marks later.

Start with the version you already write in code. Here is a test that decides whether a number belongs to a set. Run it and note which candidates get in.

The set that test describes is every number from -3 up to but not including 5. The endpoint -3 is in because the test allows equality there. The endpoint 5 is out because it does not.

Written in mathematics that test is $-3 \le x < 5$. The symbol $\le$ is read as is less than or equal to, and it is what the code writes as <=. The symbol $<$ is read as is less than, with no equality allowed, and the code writes it the same way.

Interval notation

Interval notation writes that set as a pair of endpoints in brackets. A square bracket means the endpoint is included, a round bracket means it is excluded. So the set above is written $[-3, 5)$.

The smaller number always goes on the left. There is no such interval as $(5, -3)$, and writing one is a wasted mark rather than an unusual set.

When a set runs on forever in one direction, the symbol $\infty$ (read as infinity) takes the place of an endpoint, and $-\infty$ does the same on the left. So $x \ge 3$ is written $[3, \infty)$, and $x < 0$ is written $(-\infty, 0)$.

Sets that come in two separate pieces are joined with the union symbol $\cup$, which means take everything in either piece. The domain of $\frac{1}{x-2}$ is everything except 2, which is $(-\infty, 2) \cup (2, \infty)$. Run the test for a two-piece set and watch the gap.

Set-builder notation

The other accepted way of writing a set describes it by its membership test rather than by its endpoints. Set-builder notation puts the variable, then a vertical bar meaning such that, then the condition.

Some books write a colon instead of the bar, as $\{x : -3 \le x < 5\}$, and it means exactly the same. Set-builder is the better choice when the condition is not a simple stretch, for instance $\{x \mid x \ne 2\}$. The symbol $\ne$ is read as is not equal to, so that set is every real number except 2, and it needs no union at all.

One worked example, done in full. Give the domain of $f(x) = \frac{5}{x + 3}$ in all three notations.

1. The only danger is a zero on the bottom, so ask when $x + 3 = 0$.
2. That happens at $x = -3$, so -3 is the one input to throw out.
3. As an inequality: $x \ne -3$.
4. As an interval: $(-\infty, -3) \cup (-3, \infty)$.
5. As set-builder: $\{x \mid x \ne -3\}$.