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Commutative

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Commutative

For other uses, see Commute (disambiguation).

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. The commutativity of simple operations, such as multiplication and addition of numbers, was for many years implicitly assumed and the property was not named until the 19th century when mathematics started to become formalized. By contrast, division and subtraction are not commutative.

Common uses

The commutative property (or commutative law) is a property associated with binary operations and functions. Similarly, if the commutative property holds for a pair of elements under a certain binary operation then it is said that the two elements commute under that operation.

Propositional logic

Template:Transformation rules

Rule of replacement

In standard truth-functional propositional logic, commutation,[1][2] or commutivity[3] are two valid rules of replacement. The rules allow one to transpose propositional variables within logical expressions in logical proofs. The rules are:

(P \or Q) \Leftrightarrow (Q \or P)

and

(P \and Q) \Leftrightarrow (Q \and 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 \and Q) \leftrightarrow (Q \and P)

Commutativity of disjunction

(P \or Q) \leftrightarrow (Q \or 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)

Set theory

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.[4][5][6]

Mathematical definitions

Further information: Symmetric function

The term "commutative" is used in several related senses.[7][8]

1. 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 noncommutative.

2. One says that x commutes with y under * if:

x * y = y * x \,

3. A binary function f \colon A \times A \to B is called commutative if:

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

History and etymology


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.[9][10] Euclid is known to have assumed the commutative property of multiplication in his book Elements.[11] 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,[12][13] 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 Philosophical Transactions of the Royal Society in 1844.[12]

Related properties

Associativity

Main article: Associative property

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 doesn't 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

f(x, y) = \frac{x + y}{2},

which is clearly commutative (interchanging x and y does not affect the result), but it is not associative (since, for example, f(1, f(2, 3)) = 1.75 but f(f(1, 2), 3) = 2.25).

Symmetry

Some forms of symmetry can be directly linked to commutativity. When a commutative operator is written as a binary function then the resulting function is symmetric across the line y = x. As an example, if we let a function f represent addition (a commutative operation) so that f(x,y) = x + y then f is a symmetric function, which can be seen in the image on the right.

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

Examples

Commutative operations in everyday life

  • Putting on socks resembles a commutative operation, since which sock is put on first is unimportant. Either way, the result (having both socks on), is the same.
  • The commutativity of addition is observed when paying for an item with cash. Regardless of the order the bills are handed over in, they always give the same total.

Commutative operations in mathematics

Two well-known examples of commutative binary operations:[7]

y + z = z + y \qquad\mbox{for all }y,z\in \mathbb{R}
For example 4 + 5 = 5 + 4, since both expressions equal 9.
y z = z y \qquad\mbox{for all }y,z\in \mathbb{R}
For example, 3 × 5 = 5 × 3, since both expressions equal 15.
  • Some binary truth functions are also commutative, since the truth tables for the functions are the same when one changes the order of the operands.
For example, Vpq = Vqp; Apq = Aqp; Dpq = Dqp; Epq = Eqp; Jpq = Jqp; Kpq = Kqp; Xpq = Xqp; Opq = Oqp.

Noncommutative operations in everyday life

  • Concatenation, the act of joining character strings together, is a noncommutative operation. For example
EA + T = EAT \neq TEA = T + EA
  • Washing and drying clothes resembles a noncommutative operation; washing and then drying produces a markedly different result to drying and then washing.
  • Rotating a book 90° around a vertical axis then 90° around a horizontal axis produces a different orientation than when the rotations are performed in the opposite order.
  • The twists of the Rubik's Cube are noncommutative. This can be studied using group theory.

Noncommutative operations in mathematics

Some noncommutative binary operations:[14]

  • Subtraction is noncommutative, since 0-1\neq 1-0
  • Division is noncommutative, since 1/2\neq 2/1
  • Some truth functions are noncommutative, since the truth tables for the functions are different when one changes the order of the operands.
For example, Bpq = Cqp; Cpq = Bqp; Fpq = Gqp; Gpq = Fqp; Hpq = Iqp; Ipq = Hqp; Lpq = Mqp; Mpq = Lqp.
  • Matrix multiplication is noncommutative since

\begin{bmatrix} 0 & 2 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 0 & 1 \\ 0 & 1 \end{bmatrix} \neq \begin{bmatrix} 0 & 1 \\ 0 & 1 \end{bmatrix} \cdot \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 0 & 1 \end{bmatrix}

Mathematical structures and commutativity

Non-commuting operators in quantum mechanics

Main article: Uncertainty principle

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

x{d\over dx}\psi = x\psi' \neq {d\over dx}x\psi = \psi + x\psi'

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 x-direction of a particle are represented respectively by the operators x and -i \hbar \frac{\partial}{\partial x} (where \hbar is the reduced Planck constant). This is the same example except for the constant -i \hbar, 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

Notes

References

Books

Abstract algebra theory. Covers commutativity in that context. Uses property throughout book.
Abstract algebra theory. Uses commutativity property throughout book.
Linear algebra theory. Explains commutativity in chapter 1, uses it throughout.

Articles

  • http://www.ethnomath.org/resources/lumpkin1997.pdf Lumpkin, B. (1997). The Mathematical Legacy Of Ancient Egypt - A Response To Robert Palter. Unpublished manuscript.
Article describing the mathematical ability of ancient civilizations.
  • Robins, R. Gay, and Charles C. D. Shute. 1987. The Rhind Mathematical Papyrus: An Ancient Egyptian Text. London: British Museum Publications Limited. ISBN 0-7141-0944-4
Translation and interpretation of the Rhind Mathematical Papyrus.

Online resources

  • Template:Springer
  • Krowne, Aaron, PlanetMath, Accessed 8 August 2007.
Definition of commutativity and examples of commutative operations
  • MathWorld., Accessed 8 August 2007.
Explanation of the term commute
  • PlanetMath, Accessed 8 August 2007
Examples proving some noncommutative operations
  • O'Conner, J J and Robertson, E F. MacTutor history of real numbers, Accessed 8 August 2007
Article giving the history of the real numbers
  • Cabillón, Julio and Miller, Jeff. Earliest Known Uses Of Mathematical Terms, Accessed 22 November 2008
Page covering the earliest uses of mathematical terms
  • O'Conner, J J and Robertson, E F. MacTutor biography of François Servois, Accessed 8 August 2007
Biography of Francois Servois, who first used the term

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