Commutative property


In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. 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.[1][2] A similar 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.[3]

Commutative property
StatementA binary operation is commutative if changing the order of the operands does not change the result.
Symbolic statement

Mathematical definitions Edit

A binary operation   on a set S is called commutative if[4][5]

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

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

Examples Edit

The cumulation of apples, which can be seen as an addition of natural numbers, is commutative.

Commutative operations Edit

The addition of vectors is commutative, because  .

Noncommutative operations Edit

Some noncommutative binary operations:[6]

Division, subtraction, and exponentiation Edit

Division is noncommutative, since  .

Subtraction is noncommutative, since  . However it is classified more precisely as anti-commutative, since  .

Exponentiation is noncommutative, since  . This property leads to two different "inverse" operations of exponentiation (namely, the nth-root operation and the logarithm operation), which is unlike the multiplication.[citation needed]

Truth functions Edit

Some truth functions are noncommutative, since the truth tables for the functions are different when one changes the order of the operands. For example, the truth tables for (A ⇒ B) = (¬A ∨ B) and (B ⇒ A) = (A ∨ ¬B) are

A B A ⇒ B B ⇒ A

Function composition of linear functions Edit

Function composition of linear functions from the real numbers to the real numbers is almost always noncommutative. For example, let   and  . Then




This also applies more generally for linear and affine transformations from a vector space to itself (see below for the Matrix representation).

Matrix multiplication Edit

Matrix multiplication of square matrices is almost always noncommutative, for example:


Vector product Edit

The vector product (or cross product) of two vectors in three dimensions is anti-commutative; i.e., b × a = −(a × b).

History and etymology Edit

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.[7][8] Euclid is known to have assumed the commutative property of multiplication in his book Elements.[9] 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,[1][10] 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.[2] 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.[11]

Propositional logic Edit

Rule of replacement Edit

In truth-functional propositional logic, commutation,[12][13] or commutativity[14] refer to two valid rules of replacement. The rules allow one to transpose propositional variables within logical expressions in logical proofs. The rules are:




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

Truth functional connectives Edit

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
Commutativity of disjunction
Commutativity of implication (also called the law of permutation)
Commutativity of equivalence (also called the complete commutative law of equivalence)

Set theory Edit

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.[15][16][17]

Mathematical structures and commutativity Edit

Related properties Edit

Associativity Edit

The associative property is closely related to the commutative property. The associative property of an expression containing two or more occurrences of the same operator states that the order operations are performed in does not affect the final result, as long as the order of terms does not change. In contrast, the commutative property states that the order of the terms does not affect the final result.

Most commutative operations encountered in practice are also associative. However, commutativity does not imply associativity. A counterexample is the function


which is clearly commutative (interchanging x and y does not affect the result), but it is not associative (since, for example,   but  ). More such examples may be found in commutative non-associative magmas. Furthermore, associativity does not imply commutativity either – for example multiplication of quaternions or of matrices is always associative but not always commutative.

Distributive Edit

Symmetry Edit

Graph showing the symmetry of the addition function

Some forms of symmetry can be directly linked to commutativity. When a commutative operation is written as a binary function   then this function is called a symmetric function, and its graph in three-dimensional space is symmetric across the plane  . For example, if the function f is defined as   then   is a symmetric function.

For relations, a symmetric relation is analogous to a commutative operation, in that if a relation R is symmetric, then  .

Non-commuting operators in quantum mechanics Edit

In quantum mechanics as formulated by Schrödinger, physical variables are represented by linear operators such as   (meaning multiply by  ), and  . These two operators do not commute as may be seen by considering the effect of their compositions   and   (also called products of operators) on a one-dimensional wave function  :


According to the uncertainty principle of Heisenberg, if the two operators representing a pair of variables do not commute, then that pair of variables are mutually complementary, which means they cannot be simultaneously measured or known precisely. For example, the position and the linear momentum in the  -direction of a particle are represented by the operators   and  , respectively (where   is the reduced Planck constant). This is the same example except for the constant  , so again the operators do not commute and the physical meaning is that the position and linear momentum in a given direction are complementary.

See also Edit

Notes Edit

  1. ^ a b Cabillón & Miller, Commutative and Distributive
  2. ^ a b Flood, Raymond; Rice, Adrian; Wilson, Robin, eds. (2011). Mathematics in Victorian Britain. Oxford University Press. p. 4. ISBN 9780191627941.
  3. ^ Weisstein, Eric W. "Symmetric Relation". MathWorld.
  4. ^ Krowne, p. 1
  5. ^ Weisstein, Commute, p. 1
  6. ^ Yark, p. 1
  7. ^ Lumpkin 1997, p. 11
  8. ^ Gay & Shute 1987
  9. ^ O'Conner & Robertson Real Numbers
  10. ^ O'Conner & Robertson, Servois
  11. ^ Gregory, D. F. (1840). "On the real nature of symbolical algebra". Transactions of the Royal Society of Edinburgh. 14: 208–216.
  12. ^ Moore and Parker
  13. ^ Copi & Cohen 2005
  14. ^ Hurley & Watson 2016
  15. ^ Axler 1997, p. 2
  16. ^ a b Gallian 2006, p. 34
  17. ^ Gallian 2006, pp. 26, 87
  18. ^ Gallian 2006, p. 236
  19. ^ Gallian 2006, p. 250

References Edit

Books Edit

  • Axler, Sheldon (1997). Linear Algebra Done Right, 2e. Springer. ISBN 0-387-98258-2.
    Abstract algebra theory. Covers commutativity in that context. Uses property throughout book.
  • Copi, Irving M.; Cohen, Carl (2005). Introduction to Logic (12th ed.). Prentice Hall. ISBN 9780131898349.
  • Gallian, Joseph (2006). Contemporary Abstract Algebra (6e ed.). Houghton Mifflin. ISBN 0-618-51471-6.
    Linear algebra theory. Explains commutativity in chapter 1, uses it throughout.
  • Goodman, Frederick (2003). Algebra: Abstract and Concrete, Stressing Symmetry (2e ed.). Prentice Hall. ISBN 0-13-067342-0.
    Abstract algebra theory. Uses commutativity property throughout book.
  • Hurley, Patrick J.; Watson, Lori (2016). A Concise Introduction to Logic (12th ed.). Cengage Learning. ISBN 978-1-337-51478-1.

Articles Edit

  • Lumpkin, B. (1997). "The Mathematical Legacy of Ancient Egypt – A Response To Robert Palter" (PDF) (Unpublished manuscript). Archived from the original (PDF) on 13 July 2007.
    Article describing the mathematical ability of ancient civilizations.
  • Gay, Robins R.; Shute, Charles C. D. (1987). The Rhind Mathematical Papyrus: An Ancient Egyptian Text. British Museum. ISBN 0-7141-0944-4.
    Translation and interpretation of the Rhind Mathematical Papyrus.

Online resources Edit

  • "Commutativity", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  • Krowne, Aaron, Commutative at PlanetMath., Accessed 8 August 2007.
    Definition of commutativity and examples of commutative operations
  • Weisstein, Eric W. "Commute". MathWorld., Accessed 8 August 2007.
    Explanation of the term commute
  • "Yark". Examples of non-commutative operations at PlanetMath., Accessed 8 August 2007
    Examples proving some noncommutative operations
  • O'Conner, J.J.; Robertson, E.F. "History of real numbers". MacTutor. Retrieved 8 August 2007.
    Article giving the history of the real numbers
  • Cabillón, Julio; Miller, Jeff. "Earliest Known Uses of Mathematical Terms". Retrieved 22 November 2008.
    Page covering the earliest uses of mathematical terms
  • O'Conner, J.J.; Robertson, E.F. "biography of François Servois". MacTutor. Archived from the original on 2 September 2009. Retrieved 8 August 2007.
    Biography of Francois Servois, who first used the term