What Is Set Builder Notation?
Set builder notation is a mathematical shorthand for describing a set by stating the properties that its members must satisfy. Instead of listing every element (roster notation), you describe the rule for membership. It is written as {x | condition} or {x : condition}, read as "the set of all x such that condition."
Common Set Builder Forms
Greater Than
All integers greater than a given value.
{x ∈ Z | x > a}
Between Two Values
All integers between two bounds.
{x ∈ Z | a < x < b}
Even Numbers
All even integers in a range.
{x ∈ Z | x = 2k, a ≤ x ≤ b}
Multiples
All multiples of k in a range.
{x ∈ Z | x = nk, a ≤ x ≤ b}
Set Builder vs Roster Notation
Set Builder Notation
Describes a set by a rule: {x | x is an even integer, 1 ≤ x ≤ 10}. This is concise and works well for large or infinite sets.
Roster Notation
Lists all elements explicitly: {2, 4, 6, 8, 10}. This is clear for small, finite sets but impractical for large or infinite sets.
Examples
- {x ∈ Z | 0 < x < 6} = {1, 2, 3, 4, 5}
- {x ∈ Z | x is even, 1 ≤ x ≤ 10} = {2, 4, 6, 8, 10}
- {x ∈ Z | x is a multiple of 3, 1 ≤ x ≤ 15} = {3, 6, 9, 12, 15}
- {x ∈ Z | x > 5} = {6, 7, 8, 9, ...} (infinite set)
Applications
- Defining domains and ranges of functions.
- Specifying solution sets for equations and inequalities.
- Database queries (conceptually similar to SQL WHERE clauses).
- Programming: list comprehensions in Python are inspired by set builder notation.