Комутативност

Појам комутативности се најчешће везује за бинарне математичке операције код којих редослед операнада не утиче на резултат операције. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says something like "3 + 4 = 4 + 3" or "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 and subtraction, that do not have it (for example, "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 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. A corresponding property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is symmetric as two equal mathematical objects are equal regardless of their order.

Математичке дефиниције

A binary operation $*$ on a set S is called commutative if

$$x*y=y*x\qquad {\mbox{for all }}x,y\in S.$$

An operation that does not satisfy the above property is called non-commutative.

One says that x commutes with y or that x and y commute under $*$ if

$$x*y=y*x.$$

In other words, an operation is commutative if every pair of elements commute.

A binary function $f\colon A\times A\to B$ is sometimes called commutative if

$$f(x,y)=f(y,x)\qquad {\mbox{for all }}x,y\in A.$$

Such a function is more commonly called a symmetric function.

Пример

Операција комутативности.

Рецимо да је дефинисана бинарна операција $\otimes :M^{2}\rightarrow X$ тако да за $\forall a,b\in M$ важи:

$a\otimes b=b\otimes a$

Онда је ова операција према дефиницији комутативна.

Уопштење

Овде се може направити и уопштење за $M^{n}\,$, $n\in N\land n\geq 2$. Операција $\otimes :M^{n}\rightarrow X$ је комутативна ако за сваку $(a_{1},...,a_{n})\in M^{n}$ и сваку њену пермутацију $(a_{\sigma (1)},...,a_{\sigma (n)})\,$ важи:

$(a_{1},...,a_{n})=(a_{\sigma (1)},...,a_{\sigma (n)})\,$ тј.

$a_{1}\otimes a_{2}\otimes \dots \otimes a_{n}=a_{\sigma (1)}\otimes a_{\sigma (2)}\otimes \dots \otimes a_{\sigma (n)}$

Историја и етимологија

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 Egyptians used the commutative property of multiplication to simplify computing products. Euclid is known to have assumed the commutative property of multiplication in his book Elements. 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 Servois in 1814, 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. in Duncan Farquharson Gregory's article entitled "On the real nature of symbolical algebra" published in 1840 in the Transactions of the Royal Society of Edinburgh.

Пропозициона логика

Rule of replacement

In truth-functional propositional logic, commutation, or commutativity refer to two valid rules of replacement. The rules allow one to transpose propositional variables within logical expressions in logical proofs. The rules are:

$(P\lor Q)\Leftrightarrow (Q\lor P)$

and

$(P\land Q)\Leftrightarrow (Q\land P)$

where "$\Leftrightarrow$" is a metalogical symbol representing "can be replaced in a proof with".

Truth functional connectives

Commutativity is a property of some logical connectives of truth functional propositional logic. The following logical equivalences demonstrate that commutativity is a property of particular connectives. The following are truth-functional tautologies.

Commutativity of conjunction

$(P\land Q)\leftrightarrow (Q\land P)$

Commutativity of disjunction

$(P\lor Q)\leftrightarrow (Q\lor P)$

Commutativity of implication (also called the law of permutation)

$(P\to (Q\to R))\leftrightarrow (Q\to (P\to R))$

Commutativity of equivalence (also called the complete commutative law of equivalence)

$(P\leftrightarrow Q)\leftrightarrow (Q\leftrightarrow P)$

Теорија скупова

In group and set theory, many algebraic structures are called commutative when certain operands satisfy the commutative property. In higher branches of mathematics, such as 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.

Математичке структуре и комутативност

• A commutative semigroup is a set endowed with a total, associative and commutative operation.
• If the operation additionally has an identity element, we have a commutative monoid
• An abelian group, or commutative group is a group whose group operation is commutative.
• A commutative ring is a ring whose multiplication is commutative. (Addition in a ring is always commutative.)
• In a field both addition and multiplication are commutative.

Види још

• Асоцијативност
• Антикомутативност
• Дистрибутивност

Литература

• Ayres, Frank (1965). Schaum's Outline of Modern Abstract Algebra (1st изд.). McGraw-Hill. ISBN 9780070026551. 
• Axler, Sheldon (1997). Linear Algebra Done Right, 2e. Springer. ISBN 0-387-98258-2. 
• Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic (12th изд.). Prentice Hall. ISBN 9780131898349. 
• Gallian, Joseph (2006). Contemporary Abstract Algebra (6e изд.). Houghton Mifflin. ISBN 0-618-51471-6. 
• Goodman, Frederick (2003). Algebra: Abstract and Concrete, Stressing Symmetry (2e изд.). Prentice Hall. ISBN 0-13-067342-0. 
• Hurley, Patrick J.; Watson, Lori (2016). A Concise Introduction to Logic (12th изд.). Cengage Learning. ISBN 978-1-337-51478-1. 
• Lumpkin, B. (1997). „The Mathematical Legacy Of Ancient Egypt — A Response To Robert Palter” (PDF) (Unpublished manuscript). Архивирано из оригинала (PDF) 13. 7. 2007. г. CS1 одржавање: Формат датума (веза)
• Gay, Robins R.; Shute, Charles C. D. (1987). The Rhind Mathematical Papyrus: An Ancient Egyptian Text. British Museum. ISBN 0-7141-0944-4. 
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• Krowne, Aaron, Commutative at PlanetMath.org., Accessed 8 August 2007.
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• „Yark”.  Examples of non-commutative operations at PlanetMath.org., Accessed 8 August 2007
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• Cabillón, Julio; Miller, Jeff. „Earliest Known Uses Of Mathematical Terms”. Приступљено 22. 11. 2008. CS1 одржавање: Формат датума (веза)
• O'Conner, J.J.; Robertson, E.F. „biography of François Servois”. MacTutor. Приступљено 8. 8. 2007. CS1 одржавање: Формат датума (веза)
• Brown, Frank Markham (2003), Boolean Reasoning: The Logic of Boolean Equations, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY.
• Chang, C.C. and 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.
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• Mendelson, Elliot (1964), Introduction to Mathematical Logic, D. Van Nostrand Company.
• Hofstadter, Douglas (1979). Gödel, Escher, Bach: An Eternal Golden Braid. Basic Books. ISBN 978-0-465-02656-2. 
• Klement, Kevin C. (2006), "Propositional Logic", in James Fieser and Bradley Dowden (eds.), Internet Encyclopedia of Philosophy, Eprint.
• Formal Predicate Calculus, contains a systematic formal development along the lines of Alternative calculus
• forall x: an introduction to formal logic, by P.D. Magnus, covers formal semantics and proof theory for sentential logic.
• Fraleigh, John B. (1976), A First Course in Abstract Algebra (2nd изд.), Reading: Addison-Wesley, ISBN 0-201-01984-1 
• Hall Jr., Marshall (1959), The Theory of Groups, New York: Macmillan 
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• Rotman, Joseph J. (1973), The Theory of Groups: An Introduction (2nd изд.), Boston: Allyn and Bacon 
• Attila Nagy (2001). Special Classes of Semigroups. Springer. ISBN 978-0-7923-6890-8
• Howie, John M. (1995), Fundamentals of Semigroup Theory, London Mathematical Society Monographs. New Series, 12, Oxford: Clarendon Press, ISBN 0-19-851194-9, Zbl 0835.20077 
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• Kilp, Mati; Knauer, Ulrich; Mikhalev, Alexander V. (2000), Monoids, acts and categories. With applications to wreath products and graphs. A handbook for students and researchers, de Gruyter Expositions in Mathematics, 29, Berlin: Walter de Gruyter, ISBN 3-11-015248-7, Zbl 0945.20036 
• Lothaire, M. (1997), Lothaire, M, ур., Combinatorics on words, Encyclopedia of Mathematics and Its Applications, 17, 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 (2nd изд.), Cambridge University Press, ISBN 0-521-59924-5, MR 1475463, Zbl 0874.20040, doi:10.1017/CBO9780511566097 

Спољашње везе

• Chapter 2 / Propositional Logic from Logic In Action
• Propositional sequent calculus prover on Project Nayuki. (note: implication can be input in the form !X|Y, and a sequent can be a single formula prefixed with > and having no commas)
• Propositional Logic - A Generative Grammar
• Weisstein, Eric W. „Binary Operation”. MathWorld.