Finite set

A finite set is a set admitting a bijection with

{0, 1, \dots, n - 1}

for some nonnegative integer

n

, including the empty set at

n = 0

. The number

n

is uniquely determined and is its cardinality.

Every injective map from a finite set to itself is surjective, and conversely. This follows by counting image elements: an injection cannot omit an element without reducing the size of its image. For infinite sets the property can fail, as

n \mapsto n + 1

is an injection of the natural numbers into themselves that omits zero. The bijection definition gives a precise meaning to finite without reusing the word in its own definition.

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Finiteness

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The property of being a finite set. In a finite set, an injective self-map is surjective; this implication can fail for an infinite set.

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A finite set is a collection of distinct elements that has a limited or countable number of members. In mathematical terms, a set \( S \) is defined as finite if there exists a natural number \( n \) such that the set contains exactly \( n \) elements. For example, the set \( S = \{1, 2, 3\} \) is a finite set because it contains three elements.

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