Комутативност — разлика између измена
Садржај обрисан Садржај додат
. |
|||
Ред 1:
{{Short description|Својство математичке операције}}{{рут}}
{{Правила трансформације}}
Појам '''комутативности''' се најчешће везује за [[бинарна операција|бинарне математичке операције]] код којих редослед операнада не утиче на резултат операције. It is a fundamental property of many binary operations, and many [[mathematical proof]]s depend on it. Most familiar as the name of the property that says something like {{nowrap|1="3 + 4 = 4 + 3"}} or {{nowrap|1="2 × 5 = 5 × 2"}}, the property can also be used in more advanced settings. The name is needed because there are operations, such as [[division (mathematics)|division]] and [[subtraction]], that do not have it (for example, {{nowrap|"3 − 5 ≠ 5 − 3"}}); such operations are ''not'' commutative, and so are referred to as ''noncommutative operations''. The idea that simple operations, such as the [[multiplication (mathematics)|multiplication]] and [[addition]] of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized.<ref name="Cabillón" /><ref name=Flood11 /> A corresponding property exists for [[binary relation]]s; a binary relation is said to be [[symmetric relation|symmetric]] if the relation applies regardless of the order of its operands; for example, [[Equality (mathematics)|equality]] is symmetric as two equal mathematical objects are equal regardless of their order.<ref>{{MathWorld|id=SymmetricRelation|title=Symmetric Relation}}</ref>
== Математичке дефиниције ==
A [[binary operation]] <math>*</math> on a [[Set (mathematics)|set]] ''S'' is called ''commutative'' if<ref name="Krowne, p.1">Krowne, p.1</ref><ref>Weisstein, ''Commute'', p.1</ref>
<math display="block">x * y = y * x\qquad\mbox{for all }x,y\in S.</math>
An operation that does not satisfy the above property is called ''non-commutative''.
One says that {{mvar|x}} ''commutes'' with {{math|''y''}} or that {{mvar|x}} and {{mvar|y}} ''commute'' under <math>*</math> if
<math display="block"> x * y = y * x.</math>
In other words, an operation is commutative if every pair of elements commute.
A [[binary function]] <math>f \colon A \times A \to B</math> is sometimes called ''commutative'' if
<math display="block">f(x, y) = f(y, x)\qquad\mbox{for all }x,y\in A.</math> Such a function is more commonly called a [[symmetric function]].
== Пример ==
Линија 17 ⟶ 31:
<math>a_1 \otimes a_2 \otimes \dots \otimes a_n = a_{\sigma(1)} \otimes a_{\sigma(2)} \otimes \dots \otimes a_{\sigma(n)}</math>
== Историја и етимологија ==
[[File:Commutative Word Origin.PNG|right|thumb|250px|The first known use of the term was in a French Journal published in 1814]]
Records of the implicit use of the commutative property go back to ancient times. The [[Egypt]]ians used the commutative property of [[multiplication]] to simplify computing [[Product (mathematics)|products]].<ref>{{harvnb|Lumpkin|1997|p=11}}</ref><ref>{{harvnb|Gay|Shute|1987}}</ref> [[Euclid]] is known to have assumed the commutative property of multiplication in his book [[Euclid's Elements|''Elements'']].<ref> O'Conner & Robertson ''Real Numbers''</ref> Formal uses of the commutative property arose in the late 18th and early 19th centuries, when mathematicians began to work on a theory of functions. Today the commutative property is a well-known and basic property used in most branches of mathematics.
The first recorded use of the term ''commutative'' was in a memoir by [[François-Joseph Servois|François Servois]] in 1814,<ref name="Cabillón">{{harvnb|Cabillón|Miller|loc=''Commutative and Distributive''}}</ref><ref>O'Conner & Robertson, ''Servois''</ref> which used the word ''commutatives'' when describing functions that have what is now called the commutative property. The word is a combination of the French word ''commuter'' meaning "to substitute or switch" and the suffix ''-ative'' meaning "tending to" so the word literally means "tending to substitute or switch". The term then appeared in English in 1838.<ref name=Flood11>{{cite book|title=Mathematics in Victorian Britain|editor1-first=Raymond|editor1-last=Flood|editor2-first=Adrian|editor2-last=Rice|editor3-first=Robin|editor3-last=Wilson|editor3-link=Robin Wilson (mathematician)|publisher=[[Oxford University Press]]|year=2011|url=https://books.google.com/books?id=YruifIx88AQC&pg=PA4|page=4|isbn=9780191627941}}</ref> in [[Duncan Farquharson Gregory]]'s article entitled "On the real nature of symbolical algebra" published in 1840 in the [[Royal Society of Edinburgh|Transactions of the Royal Society of Edinburgh]].<ref>{{Cite journal|first=D. F. |last=Gregory|title=On the real nature of symbolical algebra|periodical=Transactions of the Royal Society of Edinburgh|volume=14|pages=208–216|year=1840|url=https://archive.org/details/transactionsofro14royal}}</ref>
== Пропозициона логика ==
=== Rule of replacement ===
In truth-functional propositional logic, ''commutation'',<ref>Moore and Parker</ref><ref>{{harvnb|Copi|Cohen|2005}}</ref> or ''commutativity''<ref>{{harvnb|Hurley|Watson|2016}}</ref> refer to two [[Validity (logic)|valid]] [[rule of replacement|rules of replacement]]. The rules allow one to transpose [[propositional variable]]s within [[well-formed formula|logical expressions]] in [[formal proof|logical proofs]]. The rules are:
:<math>(P \lor Q) \Leftrightarrow (Q \lor P)</math>
and
:<math>(P \land Q) \Leftrightarrow (Q \land P)</math>
where "<math>\Leftrightarrow</math>" is a [[metalogic]]al [[Symbol (formal)|symbol]] representing "can be replaced in a [[Formal proof|proof]] with".
=== Truth functional connectives ===
''Commutativity'' is a property of some [[logical connective]]s of truth functional [[propositional logic]]. The following [[logical equivalence]]s demonstrate that commutativity is a property of particular connectives. The following are truth-functional [[tautology (logic)|tautologies]].
;Commutativity of conjunction:<math>(P \land Q) \leftrightarrow (Q \land P)</math>
;Commutativity of disjunction:<math>(P \lor Q) \leftrightarrow (Q \lor P)</math>
;Commutativity of implication (also called the law of permutation):<math>(P \to (Q \to R)) \leftrightarrow (Q \to (P \to R))</math>
;Commutativity of equivalence (also called the complete commutative law of equivalence):<math>(P \leftrightarrow Q) \leftrightarrow (Q \leftrightarrow P)</math>
== Теорија скупова ==
In [[group theory|group]] and [[set theory]], many algebraic structures are called commutative when certain operands satisfy the commutative property. In higher branches of mathematics, such as [[Mathematical analysis|analysis]] and [[linear algebra]] the commutativity of well-known operations (such as [[addition]] and [[multiplication]] on real and complex numbers) is often used (or implicitly assumed) in proofs.<ref>{{harvnb|Axler|1997|p=2}}</ref><ref name="Gallian, p.34">{{harvnb|Gallian|2006|p=34}}</ref><ref>{{harvnb|Gallian|2006|pp=26,87}}</ref>
== Математичке структуре и комутативност ==
* A [[commutative semigroup]] is a set endowed with a total, [[associativity|associative]] and commutative operation.<ref>[[A. H. Clifford]], [[G. B. Preston]] (1964). ''The Algebraic Theory of Semigroups Vol. I'' (Second Edition). [[American Mathematical Society]]. {{isbn|978-0-8218-0272-4}}</ref><ref>A. H. Clifford, G. B. Preston (1967). ''The Algebraic Theory of Semigroups Vol. II'' (Second Edition). [[American Mathematical Society]]. {{isbn|0-8218-0272-0}}</ref>
* If the operation additionally has an [[identity element]], we have a [[commutative monoid]]<ref>{{cite book|first1=Michel |last1=Gondran |first2=Michel |last2=Minoux |title=Graphs, Dioids and Semirings: New Models and Algorithms |year=2008 |location=Dordrecht |publisher=[[Springer-Verlag]] |isbn=978-0-387-75450-5 |zbl=1201.16038 |series=Operations Research/Computer Science Interfaces Series |volume=41 | page=13}}</ref>
* An [[abelian group]], or ''commutative group'' is a [[group (mathematics)|group]] whose group operation is commutative.<ref name="Gallian, p.34"/>
* A [[commutative ring]] is a [[ring (mathematics)|ring]] whose [[multiplication]] is commutative. (Addition in a ring is always commutative.)<ref>{{harvnb|Gallian|2006|p=236}}</ref>
* In a [[field (mathematics)|field]] both addition and multiplication are commutative.<ref>{{harvnb|Gallian|2006|p=250}}</ref>
== Види још ==
Линија 24 ⟶ 72:
* [[Антикомутативност]]
* [[Дистрибутивност]]
== Референце ==
{{reflist|}}
== Литература ==
{{refbegin|30em}}
* {{cite book|author=Ayres, Frank|title=Schaum's Outline of Modern Abstract Algebra|location=|publisher=McGraw-Hill|edition=1st|year=1965|isbn=9780070026551|pages=}}
* {{Cite book| first=Sheldon | last=Axler | title=Linear Algebra Done Right, 2e | publisher=Springer | year=1997 | isbn=0-387-98258-2}}
* {{Cite book |last1=Copi |first1=Irving M. |last2=Cohen |first2=Carl |title=Introduction to Logic |publisher=Prentice Hall |year=2005 |edition=12th |isbn=9780131898349 }}
* {{Cite book|first=Joseph|last=Gallian|title=Contemporary Abstract Algebra |edition=6e|year=2006|isbn=0-618-51471-6|publisher=Houghton Mifflin }}
* {{Cite book| first=Frederick | last=Goodman | title=Algebra: Abstract and Concrete, Stressing Symmetry |edition=2e | publisher=Prentice Hall | year=2003 | isbn=0-13-067342-0}}
* {{cite book |first1=Patrick J. |last1=Hurley |first2=Lori |last2=Watson |title=A Concise Introduction to Logic |url=https://books.google.com/books?id=l-W5DQAAQBAJ&pg=PA675 |date=2016 |publisher=Cengage Learning |isbn=978-1-337-51478-1 |edition=12th}}
* {{cite web |url=http://www.ethnomath.org/resources/lumpkin1997.pdf |last=Lumpkin |first=B. |year=1997 |title=The Mathematical Legacy Of Ancient Egypt — A Response To Robert Palter |archive-url=https://web.archive.org/web/20070713072942/http://www.ethnomath.org/resources/lumpkin1997.pdf |archive-date=13 July 2007 |type=Unpublished manuscript}}
* {{cite book |first1=Robins R. |last1=Gay |first2=Charles C. D. |last2=Shute |year=1987 |title=The Rhind Mathematical Papyrus: An Ancient Egyptian Text |publisher=British Museum |isbn=0-7141-0944-4}}
* {{springer|title=Commutativity|id=p/c023420|ref=none}}
* Krowne, Aaron, {{PlanetMath|title=Commutative|urlname=Commutative}}, Accessed 8 August 2007.
* {{MathWorld|title=Commute|urlname=Commute}}, Accessed 8 August 2007.
* {{cite web |url=http://planetmath.org/?op=getuser&id=2760 |title=Yark |ref={{harvid|Yark}}}} {{PlanetMath|title=Examples of non-commutative operations|urlname=ExampleOfCommutative}}, Accessed 8 August 2007
* {{cite web |last1=O'Conner |first1=J.J. |last2=Robertson |first2=E.F. |url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Real_numbers_1.html |title=History of real numbers |work=MacTutor |access-date=8 August 2007 }}
* {{cite web |last1=Cabillón |first1=Julio |last2=Miller |first2=Jeff |url=http://jeff560.tripod.com/c.html |title=Earliest Known Uses Of Mathematical Terms |access-date=22 November 2008}}
* {{cite web |last1=O'Conner |first1=J.J. |last2=Robertson |first2=E.F. |url=http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Servois.html |title=biography of François Servois |work=MacTutor |access-date=8 August 2007 }}
* Brown, Frank Markham (2003), ''Boolean Reasoning: The Logic of Boolean Equations'', 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY.
* [[Chen Chung Chang|Chang, C.C.]] and [[Howard Jerome Keisler|Keisler, H.J.]] (1973), ''Model Theory'', North-Holland, Amsterdam, Netherlands.
* Kohavi, Zvi (1978), ''Switching and Finite Automata Theory'', 1st edition, McGraw–Hill, 1970. 2nd edition, McGraw–Hill, 1978.
* [[Robert R. Korfhage|Korfhage, Robert R.]] (1974), ''Discrete Computational Structures'', Academic Press, New York, NY.
* [[Joachim Lambek|Lambek, J.]] and Scott, P.J. (1986), ''Introduction to Higher Order Categorical Logic'', Cambridge University Press, Cambridge, UK.
* Mendelson, Elliot (1964), ''Introduction to Mathematical Logic'', D. Van Nostrand Company.
* {{Cite book|last=Hofstadter |first=Douglas |author-link=Douglas Hofstadter |title=[[Gödel, Escher, Bach|Gödel, Escher, Bach: An Eternal Golden Braid]] |year=1979 |publisher=[[Basic Books]] |isbn=978-0-465-02656-2 }}
* [[Kevin C. Klement|Klement, Kevin C.]] (2006), "Propositional Logic", in James Fieser and Bradley Dowden (eds.), ''[[Internet Encyclopedia of Philosophy]]'', [http://www.iep.utm.edu/p/prop-log.htm Eprint].
* [http://www.qedeq.org/current/doc/math/qedeq_formal_logic_v1_en.pdf Formal Predicate Calculus], contains a systematic formal development along the lines of [[Propositional calculus#Alternative calculus|Alternative calculus]]
* ''[http://www.fecundity.com/logic/ forall x: an introduction to formal logic]'', by [[P.D. Magnus]], covers formal semantics and [[proof theory]] for sentential logic.
* {{citation|last=Fraleigh|first=John B.|title=A First Course in Abstract Algebra|edition=2nd|publisher=Addison-Wesley|place=Reading|year=1976|isbn=0-201-01984-1}}
* {{citation|last=Hall Jr.|first= Marshall|title=The Theory of Groups|publisher=Macmillan|place=New York|year=1959}}
* {{citation|last1=Hardy|first1=Darel W.|last2=Walker|first2=Carol L.|title=Applied Algebra: Codes, Ciphers and Discrete Algorithms|publisher=Prentice-Hall|place=Upper Saddle River, NJ|year=2002|isbn=0-13-067464-8}}
* {{citation|last=Rotman|first=Joseph J.|title=The Theory of Groups: An Introduction|publisher=Allyn and Bacon|place=Boston|year=1973|edition=2nd}}
* Attila Nagy (2001). ''Special Classes of Semigroups''. [[Springer Science+Business Media|Springer]]. {{isbn|978-0-7923-6890-8}}
* {{citation | first=John M. | last=Howie | author-link=John Mackintosh Howie | title=Fundamentals of Semigroup Theory | series=London Mathematical Society Monographs. New Series | volume=12 | year=1995 | publisher=Clarendon Press | location=Oxford | isbn=0-19-851194-9 | zbl=0835.20077 }}
* {{citation | last=Jacobson | first=Nathan | author-link=Nathan Jacobson | title=Lectures in Abstract Algebra | volume=I | publisher=D. Van Nostrand Company | year=1951 | isbn=0-387-90122-1}}
* {{Citation|last=Jacobson |first=Nathan |author-link=Nathan Jacobson |year=2009 |title=Basic algebra |edition=2nd |volume=1 |publisher=Dover |isbn=978-0-486-47189-1 }}
* {{citation | zbl=0945.20036 | last1=Kilp | first1=Mati | last2=Knauer | first2=Ulrich | last3=Mikhalev | first3=Alexander V. | title=Monoids, acts and categories. With applications to wreath products and graphs. A handbook for students and researchers| series=de Gruyter Expositions in Mathematics | volume=29 | location=Berlin | publisher=Walter de Gruyter | year=2000 | isbn=3-11-015248-7 }}
* {{citation | last=Lothaire | first=M. | author-link=M. Lothaire | others=Perrin, D.; Reutenauer, C.; Berstel, J.; Pin, J. E.; Pirillo, G.; Foata, D.; Sakarovitch, J.; Simon, I.; Schützenberger, M. P.; Choffrut, C.; Cori, R.; Lyndon, Roger; Rota, Gian-Carlo. Foreword by Roger Lyndon | editor1-first=M | editor1-last=Lothaire | title=Combinatorics on words | edition=2nd | series=Encyclopedia of Mathematics and Its Applications | volume=17 | publisher=[[Cambridge University Press]] | year=1997 | doi = 10.1017/CBO9780511566097 | isbn=0-521-59924-5 | mr = 1475463 | zbl=0874.20040 }}
{{refend}}
== Спољашње везе ==
{{Commons category-lat|Commutative property}}
* [http://logicinaction.org/docs/ch2.pdf Chapter 2 / Propositional Logic] from [http://logicinaction.org Logic In Action]
* [https://www.nayuki.io/page/propositional-sequent-calculus-prover Propositional sequent calculus prover] on Project Nayuki. (''note'': implication can be input in the form <tt>!X|Y</tt>, and a sequent can be a single formula prefixed with <tt>></tt> and having no commas)
* [https://docs.google.com/document/d/1DhtRAPcMwJmiQnbdmFcHWaOddQ7kuqqDnWp2LZcGlnY/edit?usp=sharing Propositional Logic - A Generative Grammar]
* {{MathWorld|title=Binary Operation|urlname=BinaryOperation}}
{{нормативна контрола}}
| |||