The cardinality of a set is the number of distinct elements it contains, written |A| or n(A). Cardinality is the foundation of counting, combinatorics, and the surprising mathematics of infinity.
Definition
The cardinality of a set is the number of distinct elements it contains, written |A| or n(A).
💡 Intuition
Cardinality answers "how many?" — count each distinct element once and you have the cardinality.
🎯 Core Idea
Cardinality measures the size of a set; infinite sets can have different cardinalities (e.g., |\mathbb{N}| < |\mathbb{R}|).
Example
Formula
Notation
|A| or n(A)
🌟 Why It Matters
Cardinality is the foundation of counting, combinatorics, and the surprising mathematics of infinity.
💭 Hint When Stuck
Write out the distinct elements, then count them. For union problems, use |A| + |B| - |A intersect B| to avoid double-counting.
Formal View
🚧 Common Stuck Point
Some infinities are bigger than others: |\text{integers}| < |\text{reals}|.
⚠️ Common Mistakes
- Counting duplicate listings — |\{1, 1, 2\}| = 2, not 3, because duplicates are collapsed in a set
- Confusing cardinality with the set itself — |\{a, b, c\}| = 3 is a number, not a set
- Assuming all infinite sets have the same cardinality — |\mathbb{N}| < |\mathbb{R}| (Cantor's theorem)
Go Deeper
Frequently Asked Questions
What is Cardinality in Math?
The cardinality of a set is the number of distinct elements it contains, written |A| or n(A).
Why is Cardinality important?
Cardinality is the foundation of counting, combinatorics, and the surprising mathematics of infinity.
What do students usually get wrong about Cardinality?
Some infinities are bigger than others: |\text{integers}| < |\text{reals}|.
What should I learn before Cardinality?
Before studying Cardinality, you should understand: set, element.