Visual Tools
Calculators
Tables
Mathematical Keyboard
Converters
Other Tools


Cardinality of Sets



Definition of Cardinality
Finite Sets
Infinite Sets
Countable Sets
Uncountable Sets
Comparing Cardinalities



Measuring the Size of Sets


Cardinality measures the size of a set — how many elements it contains. For finite sets, cardinality is simply a count. For infinite sets, the situation becomes more subtle: different infinite sets can have different sizes, and comparing them requires the concept of one-to-one correspondence. This leads to the distinction between countable and uncountable infinities.



Definition of Cardinality


The cardinality of a set AA, denoted ∣A∣|A|, measures the number of elements in AA.

For finite sets, cardinality is determined by counting. If A={a,b,c}A = \{a, b, c\}, then ∣A∣=3|A| = 3. If B={1,2,3,4,5}B = \{1, 2, 3, 4, 5\}, then ∣B∣=5|B| = 5.

The empty set contains no elements:

∣∅∣=0|\emptyset| = 0


Two sets have the same cardinality when a bijection (one-to-one correspondence) exists between them. For finite sets, this simply means they have the same count. For infinite sets, this definition becomes the primary tool for comparing sizes.

Finite Sets


    A set is finite if its elements can be counted with a natural number. Formally, a set AA is finite if there exists some n∈Nn \in \mathbb{N} such that:

    ∣A∣=n|A| = n


    This means a bijection exists between AA and the set {1,2,3,…,n}\{1, 2, 3, \ldots, n\}.

    Examples of finite sets:

  • 2626

  • 2020 is {2,3,5,7,11,13,17,19}\{2, 3, 5, 7, 11, 13, 17, 19\} with cardinality 88

  • subset of a finite set is finite

    • The cardinality of a finite set satisfies:

    ∣A∪B∣=∣A∣+∣B∣−∣A∩B∣|A \cup B| = |A| + |B| - |A \cap B|


    This inclusion-exclusion principle accounts for elements counted twice when sets overlap.

Infinite Sets


    A set is infinite if it is not finite — no natural number can express its size.

    The standard number sets are all infinite:

  • N={0,1,2,3,…}\mathbb{N} = \{0, 1, 2, 3, \ldots\} — the natural numbers

  • Z={…,−2,−1,0,1,2,…}\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\} — the integers

  • Q\mathbb{Q} — the rational numbers

  • R\mathbb{R} — the real numbers

    • A defining property of infinite sets: an infinite set can be put into one-to-one correspondence with a proper subset of itself. For example, the function f(n)=2nf(n) = 2n maps N\mathbb{N} bijectively onto the even numbers, which form a proper subset of N\mathbb{N}.

    Not all infinite sets have the same cardinality. The integers and rationals turn out to be the same size as the natural numbers, but the real numbers form a strictly larger infinity.

Countable Sets


A set is countably infinite if its elements can be put in one-to-one correspondence with the natural numbers N\mathbb{N}. Such a set can be listed as a sequence:

a1,a2,a3,…a_1, a_2, a_3, \ldots


where every element appears exactly once.

The integers Z\mathbb{Z} are countable. The listing:

0,1,−1,2,−2,3,−3,…0, 1, -1, 2, -2, 3, -3, \ldots


establishes a bijection with N\mathbb{N}.

The rationals Q\mathbb{Q} are countable despite appearing denser than Z\mathbb{Z}. By arranging positive rationals in a grid and traversing diagonally, every rational is eventually listed.

A set is called countable if it is either finite or countably infinite. Some authors reserve "countable" for countably infinite sets only — context determines the convention.

The union of countably many countable sets remains countable. This is why Q\mathbb{Q} is countable: it is a countable union of countable sets (rationals with each fixed denominator).

Uncountable Sets


A set is uncountable if it is infinite but not countable — its elements cannot be listed in a sequence that covers them all.

The real numbers R\mathbb{R} are uncountable. Cantor's diagonal argument proves this: assume a listing of all real numbers between 00 and 11 exists. Construct a new number by making its nn-th decimal digit different from the nn-th digit of the nn-th number in the list. This new number differs from every listed number, contradicting the assumption that the list was complete.

Any interval (a,b)(a, b) with a<ba < b is uncountable. In fact, every such interval has the same cardinality as R\mathbb{R} itself — a bijection exists between them.

The cardinality of R\mathbb{R} is denoted c\mathfrak{c} (for continuum) or 2ℵ02^{\aleph_0}, indicating it equals the cardinality of the power set of N\mathbb{N}.

Comparing Cardinalities


    Cardinalities are compared using functions between sets:

  • ∣A∣=∣B∣|A| = |B| when a bijection exists between AA and BB

  • ∣A∣≤∣B∣|A| \leq |B| when an injection (one-to-one function) exists from AA to BB

  • ∣A∣<∣B∣|A| < |B| when ∣A∣≤∣B∣|A| \leq |B| and ∣A∣≠∣B∣|A| \neq |B|

    • The Cantor-Schröder-Bernstein theorem states: if ∣A∣≤∣B∣|A| \leq |B| and ∣B∣≤∣A∣|B| \leq |A|, then ∣A∣=∣B∣|A| = |B|.

    Cantor's theorem establishes a fundamental inequality for any set AA:

    ∣A∣<∣P(A)∣|A| < |\mathcal{P}(A)|


    The power set always has strictly greater cardinality than the original set. For finite sets, this is immediate: 2n>n2^n > n. For infinite sets, the proof uses a diagonal argument similar to the uncountability of R\mathbb{R}.

    This theorem implies there is no largest cardinality — given any set, its power set is strictly larger, producing an endless hierarchy of infinite sizes.