Cardinality
Calculus & Advanced MathCardinality is the measure of the number of elements in a set, extending the concept of "size" to infinite sets.
Formula
|A| = \text{number of elements in } A
Definition
Cardinality is just the count of how many elements are in a set. It is the "size" of a set.
Example
$|\{a, b, c\}| = 3$. $|\{1, 2, \ldots, 100\}| = 100$. $|\emptyset| = 0$. You count the elements.
Key Insight
For finite sets, cardinality is simply counting. The concept becomes fascinating and strange for infinite sets.
Definition
Two sets have the same cardinality if there is a bijection (one-to-one correspondence) between them. For finite sets this matches ordinary counting. $|\mathbb{N}| = |\mathbb{Z}| = |\mathbb{Q}|$ (all countably infinite, denoted $\aleph_0$), but $|\mathbb{R}| > |\mathbb{N}|$ (uncountable).
Example
The bijection $n \mapsto 2n$ matches each natural number to an even natural number, so $|\mathbb{N}| = |\text{even naturals}|$, a surprising result for infinite sets.
Key Insight
Cantor's diagonal argument proves $|\mathbb{R}| > |\mathbb{N}|$: no list can exhaust all real numbers, so reals are "more infinite" than naturals.
Definition
Cardinality is an equivalence class of sets under bijection. Cantor-Bernstein-Schroeder theorem: if $|A| \le |B|$ and $|B| \le |A|$ then $|A| = |B|$. The cardinal hierarchy: $\aleph_0, \aleph_1, \ldots$ The Continuum Hypothesis ($|\mathbb{R}| = \aleph_1$) is independent of ZFC.
Example
$|\mathbb{R}| = |\mathbb{R}^2| = |\mathbb{R}^n|$ for all finite $n$: the plane and the line have the same cardinality. This shocked Cantor himself.
Key Insight
The independence of the Continuum Hypothesis (proved by Godel and Cohen) shows there are genuinely irresolvable questions in mathematics, even about something as basic as the size of the real numbers.