Concept

Finite set

Overview

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,

{

2

,

4

,

6

,

8

,

10

}

{\displaystyle \{2,4,6,8,10\}}

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

{

1

,

2

,

3

,

…

}

{\displaystyle \{1,2,3,\ldots \}}

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.

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

Possible passages

Found by search and not yet checked by a person; they may only mention the idea in passing.

1. Thus there arises a classification of modes of formation, or in the language commonly used by mathematicians, a classification of Functions. Any number, considered as formed from any other number, is called a function of it; and there are as many kinds of functions as there are modes of formation. The simple functions are by no means numerous, most functions being formed by the combination of several of the…

   John Stuart Mill, A System of Logic, Ratiocinative and Inductive

   States what Finite set is.

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2. All types of series are capable of formal definition, and their properties can be deduced from the principles of symbolic logic by means of the Algebra of Relatives. The notion of a limit, which is fundamental in the greater part of higher mathematics, used to be defined by means of quantity, as a term to which the terms of some series approximate as nearly as we please. But nowadays the limit is defined quite…

   Bertrand Russell, Mysticism and Logic and Other Essays

   States what Finite set is.

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3. It must be granted that Cantor’s definition includes every series that is continuous; nor can it be objected that it includes any important or indubitable case of a series not continuous. Nevertheless, it has some serious defects. In the first place, it turns upon metrical considerations; while the distinction between a continuous and a discontinuous series is manifestly non-metrical. In the next place, a perfect…

   Charles S. Peirce, Chance, Love, and Logic

   States what Finite set is.

   Read the chapter · Open in the reader

Related ideas

• Infinite set
• Zermelo–Fraenkel set theory
• Function (mathematics)
• Mathematics
• Mathematical induction
• Mathematical object
• Model theory
• Natural number
• Richard Dedekind
• Power set
• Peano axioms
• Set theory
• Subset
• Total order
• Bijection
• Kazimierz Kuratowski
• Union (set theory)
• Combinatorics
• Alfred Tarski
• Cardinal number
• Cardinality
• Element of a set
• Mathematical proof
• Axiom of choice

Opposing perspectives

• Sequence

Place in the tradition

Part of Fallibilism.

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