Translating Verbal Conditions into Set and Interval Notation

Hello! This first module begins with sets: a way to describe exactly which objects belong to a collection. We will use them later to study relations, functions, and mathematical logic. Today’s task is practical: take a verbal condition and write the same set as a roster, in set-builder notation, or—when it describes a range of real numbers—in interval notation.

The key habit is to ask two questions before choosing symbols: What kind of numbers are allowed? and Which of those numbers satisfy the condition?

Rosters: show the members

A set needs a clear membership rule. “The integers greater than 2 and less than 7” is precise; “some fairly large numbers” is not. In roster form, you list the members inside curly braces:

Order does not matter, and repeating a member changes nothing: $\{3,4,5\}$ and $\{5,3,3,4\}$ describe the same set. We normally write each member just once. The symbol $x\in A$ means “$x$ belongs to $A$.”

Watch the roster and set-builder examples in Roster and Set-builder form, from Khan Academy India – English. Notice when the speaker lists answers and when they state a rule for finding them.

Number type matters immediately. Throughout this lesson, $\mathbb{N}=\{1,2,3,\dots \}$ denotes the natural numbers, $\mathbb{Z}$ the integers (including zero and negative integers), and $\mathbb{R}$ the real numbers.

Consider “the natural numbers that are at least 2, even, and less than 12.” Checking those conditions gives

For a larger patterned set, an ellipsis is useful if the continuation is unmistakable: $\{2,4,6,\dots \}$ means the positive even integers without an upper limit. An ellipsis is not a substitute for a clear rule when several continuations are possible.

A roster can also be empty. There is no natural number strictly between 0 and 1, so its roster is $\varnothing$, also written $\{\}$. That is not $\{0\}$, which contains one number.

Set-builder notation: state the membership test

A roster gives the members; set-builder notation gives a test that every member must pass. A useful template is

Read this as “the set of $x$ in $D$ such that the condition holds.” The symbol $\mid$ means such that; a colon is often used instead. The set $D$ specifies the allowed type of object, sometimes called the domain of discussion.

For the even-number example above, we can write

Both restrictions matter. If you omit “even,” the set also contains 3, 5, and other odd numbers. If you omit $\mathbb{N}$, the intended kind of number is less explicit.

Here is a condition that requires more work than simply counting along a number line: “the integers whose squares are less than 10.” Set-builder form preserves the statement directly:

Since $3^2=9<10$ but $4^2=16>10$, the permissible integers run from $-3$ through $3$. Thus

The negative values are easy to miss if you start listing before examining the condition. Notice also that changing “integers” to “natural numbers” would change the roster, even though $x^2<10$ stayed the same.

Intervals: include every real number in a range

Now change the wording to “real numbers greater than 1 and less than 5.” A roster such as $\{2,3,4\}$ misses $1.5$, $2.01$, and infinitely many other members. Interval notation is designed for a continuous range of real numbers:

A parenthesis excludes an endpoint; a square bracket includes it. Watch Intervals and interval notation, from Khan Academy, as the number-line drawings are translated into inequalities and intervals.

For example, “real numbers at least $-2$ but below $4$” includes $-2$ and excludes $4$:

Translate each end separately. “At least,” “at most,” “no less than,” and “no more than” permit equality; “greater than,” “below,” and “strictly between” do not. If a prompt just says “between,” check whether it specifies the endpoints—the word alone can be ambiguous.

Choose the notation that preserves the number type

Intervals can extend indefinitely. In the final part of the interval video, pay attention to why infinity always gets a parenthesis and why excluding one point splits a range into two pieces.

For example, “real numbers at least 3” is

Infinity is not a real endpoint that the set can contain, so it never receives a square bracket. Similarly, “all real numbers except 1” requires two intervals:

Here $\cup$ joins the two pieces. Set operations, including this one, are the subject of the next lesson.

An interval describes real numbers, not just the integers within its bounds. Compare these two conditions:

The second set contains $2.5$; the first does not. Consequently, $(2,7]$ is not interval notation for the first set.

When translating a new description, work in this order:

1. Identify the allowed objects: natural numbers, integers, or real numbers?
2. Turn every verbal restriction into a membership condition, paying special attention to equality at boundaries.
3. Choose a roster for a manageable list, set-builder notation for a precise rule, or intervals for real-number ranges. Check an endpoint and a value inside the range against your result.

The central idea is that notation must preserve both the condition and the kind of number allowed. Rosters display members, set-builder notation states their membership test, and intervals capture real-number ranges with precise endpoints. Next, you will use these sets to work with unions, intersections, differences, and complements.