          # Finite set

• in mathematics, a finite set is a set that has a finite number of elements. 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 (a non-negative integer) and is called the cardinality of the set. a set that is not finite is called infinite. 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
• basic properties
• necessary and sufficient conditions for finiteness
• foundational issues
• set-theoretic definitions of finiteness
## In mathematics, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, $\{2,4,6,8,10\}$ is a finite set with five elements. The number of elements of a finite set is a natural number (a non-negative integer) and is called the cardinality of the set. A set that is not finite is called infinite. For example, the set of all positive integers is infinite: $\{1,2,3,\ldots \}.$ 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. Contents 1 Definition and terminology 2 Basic properties 3 Necessary and sufficient conditions for finiteness 4 Foundational issues 5 Set-theoretic definitions of finiteness 5.1 Other concepts of finiteness 6 See also 7 Notes 8 References 9 External links  