The cardinality of a finite set is the number of distinct elements it contains, written |A| — it measures the size of the set without regard to element order or identity. Cardinality is the foundation of counting, combinatorics, and the surprising mathematics of infinity.
Definition
The cardinality of a finite set is the number of distinct elements it contains, written |A| — it measures the size of the set without regard to element order or identity.
💡 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 finite set is the number of distinct elements it contains, written |A| — it measures the size of the set without regard to element order or identity.
What is the Cardinality formula?
|A \cup B| = |A| + |B| - |A \cap B| (inclusion-exclusion principle)
When do you use Cardinality?
Write out the distinct elements, then count them. For union problems, use |A| + |B| - |A intersect B| to avoid double-counting.