How Do You List the Elements of a Set?

To list the elements of a set, you need to know how the set is defined. The title alone does not specify a set, so there is no particular list of elements to give. If you have a set expression or a rule for membership, you can expand it using the method below.

What it means to list a set

A set is a collection of distinct objects. When you write its elements explicitly, put them inside braces and separate them with commas. For example, the set of positive integers less than 5 is

$$\{1,2,3,4\}.$$

The order does not matter: $\{1,2,3,4\}$ and $\{4,2,1,3\}$ represent the same set. Repeated elements do not change a set either, so $\{1,2,2,3\}=\{1,2,3\}$.

A rule may describe a set without listing its elements. This is often called set-builder notation. For example,

$$\{x\in\mathbb{Z}: 0<x<5\}$$

means “the integers $x$ such that $0<x<5$.” The integers satisfying the rule are $1,2,3,4$, so the set can be listed as $\{1,2,3,4\}$.

A step-by-step method

1. Read the domain. Find out what kinds of objects are allowed. The condition $x^2=4$ has different answers depending on whether $x$ must be an integer, a real number, or another kind of object.
2. Apply the membership rule. Determine which allowed objects satisfy every condition in the definition.
3. Check for omitted cases. In a finite range, test all candidates. With a divisibility or parity rule, check each candidate against that rule.
4. Write distinct elements in braces. For a finite set, include every qualifying element once. For an infinite set, make clear that the pattern continues or retain the defining rule.

Worked examples

Example 1: A finite set defined by a rule. List the elements of

$$A=\{n\in\mathbb{Z}: -2\le n<3\}.$$

The domain is the integers. The values from $-2$ through $2$ satisfy the inequalities, while $3$ does not because the upper bound is strict. Therefore,

$$A=\{-2,-1,0,1,2\}.$$

Example 2: A set defined by divisibility. List the positive integers less than 20 that are divisible by 4. Check the positive multiples of 4 below 20: $4,8,12,16$. The next multiple, 20, is excluded. The set is

$$\{4,8,12,16\}.$$

Example 3: An infinite set. The positive even integers can be written

$$\{2,4,6,8,\ldots\}.$$

The ellipsis means the displayed pattern continues; it does not indicate that 8 is the last element. To avoid relying on a pattern, write the set-builder form $\{2n:n\in\mathbb{Z},\ n>0\}$.

Common mistakes

• Leaving out the domain: $\{x:x^2=4\}$ is not fully specified unless the universe for $x$ is understood. Over the integers or reals, its elements are $-2$ and $2$; over the natural numbers, whether 0 is included makes no difference here: the only solution is $2$.
• Including a strict endpoint: The condition $x<5$ excludes 5; $x\le 5$ includes it.
• Listing duplicates: A set contains each element only once, even if the defining process produces the same value multiple times.
• Treating an ellipsis as a complete finite list: Use an ellipsis only when the continuing pattern is clear, and say if the set is infinite.
• Confusing an element with a subset: For instance, $\{1,2\}$ has elements 1 and 2. The set $\{1,\{2\}\}$ has elements 1 and the set $\{2\}$.

To determine a particular list, provide the set itself—such as a roster, a set-builder rule, or a description of its domain and membership condition.