Cardinality
Elementary School
Definition
This is the number of elements that you have in a set. In a finite set, there is a value for cardinality which helps you determine the set's size. Cardinality is denoted by |A|. You can say that |A| = 4 for a set that has four elements in it.
Worked examples
\(A = \{2, 4, 6, 8\}\) so \(|A| = 4\)
Count the elements in the set; there are four, so the cardinality is 4.
\(B = \{x, y, x, z\} = \{x, y, z\}\) so \(|B| = 3\)
Duplicates are removed in a set, so count only the distinct elements.
\(C = \{\}\) so \(|C| = 0\)
The empty set has no elements, so its cardinality is zero.
Common mistakes
- \(|\{1, 2, 2, 3\}| = 4\) → \(|\{1, 2, 2, 3\}| = |\{1, 2, 3\}| = 3\) Sets contain only distinct elements; duplicates are not counted separately.
- \(|\{5\}| = 5\) → \(|\{5\}| = 1\) Cardinality counts how many elements are in the set, not the value of the elements.
- Writing \(|A|\) when \(A\) is not defined as a set → Define \(A\) first, e.g. \(A = \{1, 3, 5\}\), then \(|A| = 3\) Cardinality notation requires a defined set; you cannot take the cardinality of nothing.
Where you'll use it next
Cardinality is essential when you count outcomes in probability, compare set sizes in Venn diagrams, and work with functions (domain and range sizes). It also appears in combinatorics and later in discrete math.
Found in 1 StudyPug lesson
Cardinal Numbers: The Foundation of Counting and Mathematics
5th Grade5thGrade 5 Math placeholder
Dive into the world of cardinal numbers! Learn how these fundamental counting numbers form the basis of mathematics and everyday calculations. Master counting, quantity, and basic arithmetic with our comprehensive guide.
See also
Reviewed by Pat Cheng, M.Ed. — StudyPug Curriculum Lead · Last updated June 6, 2026