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Cardinality

Cardinality of a Set

The number of elements in a set., whether the set is finite or infinite. Note: Not all infinite sets have the same cardinality.

 

 

See also

Countably infinite, uncountably infinite, cardinal numbers, infinity

Key Formula

∣A∣=n|A| = n
Where:
  • ∣A∣|A| = The cardinality of set A
  • nn = The number of elements in set A (a non-negative integer for finite sets, or a cardinal number for infinite sets)

Worked Example

Problem: Find the cardinality of A = {2, 5, 7, 10, 13}.
Step 1: List every distinct element in the set.
A={2, 5, 7, 10, 13}A = \{2,\, 5,\, 7,\, 10,\, 13\}
Step 2: Count the elements. There are 5 distinct elements.
∣A∣=5|A| = 5
Answer: The cardinality of A is 5.

Another Example

Problem: Find the cardinality of B = {a, b, c, a, b}.
Step 1: A set contains only distinct elements. Remove duplicates.
B={a, b, c}B = \{a,\, b,\, c\}
Step 2: Count the distinct elements.
∣B∣=3|B| = 3
Answer: The cardinality of B is 3, because duplicates are not counted separately in a set.

Frequently Asked Questions

Do all infinite sets have the same cardinality?
No. The set of natural numbers and the set of integers are both countably infinite and share the same cardinality, denoted ℵ₀ (aleph-null). However, the set of real numbers is uncountably infinite and has a strictly larger cardinality. Georg Cantor proved this with his famous diagonal argument.
What is the cardinality of the empty set?
The empty set ∅ has no elements, so its cardinality is 0: |∅| = 0.

Countably Infinite vs. Uncountably Infinite

A set is countably infinite if its elements can be put into a one-to-one correspondence with the natural numbers (e.g., the integers or the rationals). Its cardinality is ℵ₀. A set is uncountably infinite if no such correspondence exists (e.g., the real numbers). Uncountable sets have a strictly greater cardinality than countable ones. Both are infinite, but they represent fundamentally different sizes of infinity.

Why It Matters

Cardinality gives you a precise way to talk about the size of any collection, whether it has 3 elements or infinitely many. It is foundational in set theory, probability (where you must know whether a sample space is finite, countable, or uncountable), and computer science (where data structures rely on knowing how many elements they hold). The discovery that different infinities exist was one of the most surprising results in the history of mathematics.

Common Mistakes

Mistake: Counting duplicate entries when finding the cardinality of a set.
Correction: A set by definition contains only distinct elements. Before counting, remove all duplicates. For example, {1, 2, 2, 3} as a set is {1, 2, 3}, so its cardinality is 3, not 4.
Mistake: Assuming all infinite sets are the same size.
Correction: The set of natural numbers (countably infinite) and the set of real numbers (uncountably infinite) have different cardinalities. Cantor's diagonal argument proves that no list can contain every real number, so the reals form a strictly larger set.

Related Terms

  • Set — The collection whose size cardinality measures
  • Element of a Set — Each item counted when finding cardinality
  • Finite — A set with a whole-number cardinality
  • Infinite — A set whose cardinality is not a finite number
  • Countably Infinite — Infinite sets with cardinality ℵ₀
  • Uncountable — Infinite sets with cardinality greater than ℵ₀
  • Cardinal Numbers — Numbers used to express cardinality
  • Infinity — Concept underlying infinite cardinalities