Finite set explained

In mathematics, a finite set is a collection of finitely many different things; the things are called elements or members of the set and are typically mathematical objects, such as numbers, symbols, points in space, lines, other geometric shapes, variables, or other sets.

Informally, a finite set is a set which one could in principle count and finish counting. For example, $\$ is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the cardinality (or the cardinal number) of the set. A set that is not a finite set is called an infinite set. For example, the set $\$ of all positive integers is infinite.

Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set.

Definition and terminology

The natural numbers are defined abstractly by the Peano axioms, and can be constructed set-theoretically (for example, by the Von Neumann ordinals). Then, formally, a set

is called finite if there exists a bijection

f\colon S\to \

for some natural number

, analogous to counting its elements. If

is empty, this is vacuously satisfied for

n=0

with the empty function. The number

is the set's cardinality, denoted as

|S|

If a nonempty set is finite, its elements may be written in a sequence: $x_1,x_2,\ldots,x_n \quad (x_i \in S, \ 1 \le i \le n).$ If n ≥ 2, then there are multiple such sequences.In combinatorics, a finite set with

elements is sometimes called an n

-set and a subset with

elements is called a k

-subset. For example, the set

\{5,6,7\}

is a 3-set – a finite set with three elements – and

\{6,7\}

is a 2-subset of it.

This notation

\{1, … ,n\}

may be defined recursively as

$\ = \left\$