# زمرة محدودة

في الرياضيات، الزمرة المحدودة Finite set هي زمرة ذات عدد عناصر محدود ، بعض نواحي الزمر المحدودة تتم معالجته بعمق في النظرية المحلية.

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## التعريف والمصطلحات

Formally, a set S is called finite if there exists a bijection

${\displaystyle f\colon S\to \{1,\ldots ,n\}}$

for some natural number n. The number n is the set's cardinality, denoted as |S|. The empty set { } or ∅ is considered finite, with cardinality zero.[1][2][3][4]

If a set is finite, its elements may be written — in many ways — in a sequence:

${\displaystyle x_{1},x_{2},\ldots ,x_{n}\quad (x_{i}\in S,\ 1\leq i\leq n).}$

In combinatorics, a finite set with n elements is sometimes called an n-set and a subset with k 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.

(Those familiar with the definition of the natural numbers themselves as conventional in set theory, the so-called von Neumann construction, may prefer to use the existence of the bijection ${\displaystyle f\colon S\to n}$ , which is equivalent.)

## الخصائص الأساسية

Any proper subset of a finite set S is finite and has fewer elements than S itself. As a consequence, there cannot exist a bijection between a finite set S and a proper subset of S. Any set with this property is called Dedekind-finite. Using the standard ZFC axioms for set theory, every Dedekind-finite set is also finite, but this implication cannot be proved in ZF (Zermelo–Fraenkel axioms without the axiom of choice) alone. The axiom of countable choice, a weak version of the axiom of choice, is sufficient to prove this equivalence.

Any injective function between two finite sets of the same cardinality is also a surjective function (a surjection). Similarly, any surjection between two finite sets of the same cardinality is also an injection.

The union of two finite sets is finite, with

${\displaystyle |S\cup T|\leq |S|+|T|.}$

In fact, by the inclusion–exclusion principle:

${\displaystyle |S\cup T|=|S|+|T|-|S\cap T|.}$

More generally, the union of any finite number of finite sets is finite. The Cartesian product of finite sets is also finite, with:

${\displaystyle |S\times T|=|S|\times |T|.}$

Similarly, the Cartesian product of finitely many finite sets is finite. A finite set with n elements has 2n  distinct subsets. That is, the power set P(S) of a finite set S is finite, with cardinality 2|S| .

Any subset of a finite set is finite. The set of values of a function when applied to elements of a finite set is finite.

All finite sets are countable, but not all countable sets are finite. (Some authors, however, use "countable" to mean "countably infinite", so do not consider finite sets to be countable.)

The free semilattice over a finite set is the set of its non-empty subsets, with the join operation being given by set union.

## الهامش

1. ^ Apostol (1974, p. 38)
2. ^ Cohn (1981, p. 7)
3. ^ Labarre, Jr. (1968, p. 41)
4. ^ Rudin (1976, p. 25)

## المراجع

• Apostol, Tom M. (1974), Mathematical Analysis (2nd ed.), Menlo Park: Addison-Wesley
• Cohn, Paul Moritz, F.R.S. (1981), Universal Algebra, Dordrecht: D. Reidel, ISBN 90-277-1254-9
• Dedekind, Richard (2012), Was sind und was sollen die Zahlen?, Cambridge Library Collection (Paperback ed.), Cambridge, UK: Cambridge University Press, ISBN 978-1-108-05038-8
• Dedekind, Richard (1963), Essays on the Theory of Numbers, Dover Books on Mathematics, Beman, Wooster Woodruff (Paperback ed.), Dover Publications Inc., ISBN 0-486-21010-3
• Herrlich, Horst (2006), Axiom of Choice, Lecture Notes in Math. 1876, Berlin: Springer-Verlag, ISBN 3-540-30989-6
• Howard, Paul; Rubin, Jean E. (1998). Consequences of the axiom of choice. Providence, Rhode Island: American Mathematical Society. ISBN 9780821809778.
• Kuratowski, Kazimierz (1920), "Sur la notion d'ensemble fini", Fundamenta Mathematicae 1: 129–131, doi:10.4064/fm-1-1-129-131
• Labarre, Jr., Anthony E. (1968), Intermediate Mathematical Analysis, New York: Holt, Rinehart and Winston
• Lévy, Azriel (1958). "The independence of various definitions of finiteness" (PDF). Fundamenta Mathematicae. 46: 1–13. doi:10.4064/fm-46-1-1-13.
• Rudin, Walter (1976), Principles Of Mathematical Analysis (3rd ed.), New York: McGraw-Hill, ISBN 0-07-054235-X
• Suppes, Patrick (1972), Axiomatic Set Theory, Dover Books on Mathematics (Paperback ed.), Dover Publications Inc., ISBN 0-486-61630-4
• Tarski, Alfred (1924). "Sur les ensembles finis" (PDF). Fundamenta Mathematicae. 6: 45–95. doi:10.4064/fm-6-1-45-95.
• Tarski, Alfred (1954). "Theorems on the existence of successors of cardinals, and the axiom of choice". Nederl. Akad. Wetensch. Proc. Ser. A., Indagationes Math. 16: 26–32. doi:10.1016/S1385-7258(54)50005-3. MR 0060555.
• Whitehead, Alfred North; Russell, Bertrand (February 2009) [1912]. Principia Mathematica. Vol. Two. Merchant Books. ISBN 978-1-60386-183-0.

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