# Commutative property

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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

### Rule of replacement

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

${\displaystyle (P\lor Q)\Leftrightarrow (Q\lor P)}$

and

${\displaystyle (P\land Q)\Leftrightarrow (Q\land P)}$

where "${\displaystyle \Leftrightarrow }$" 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

${\displaystyle (P\land Q)\leftrightarrow (Q\land P)}$

Commutativity of disjunction

${\displaystyle (P\lor Q)\leftrightarrow (Q\lor P)}$

Commutativity of implication (also called the Law of permutation)

${\displaystyle (P\to (Q\to R))\leftrightarrow (Q\to (P\to R))}$

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

${\displaystyle (P\leftrightarrow Q)\leftrightarrow (Q\leftrightarrow P)}$

## 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

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

1. A binary operation ${\displaystyle *}$ on a set S is called commutative if:

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

An operation that does not satisfy the above property is called noncommutative.

2. One says that x commutes with y under ${\displaystyle *}$ if:

${\displaystyle x*y=y*x\,}$

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

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

## History and etymology

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.[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

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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

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

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

### Symmetry

Graph showing the symmetry of the addition function

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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 ${\displaystyle aRb\Leftrightarrow bRa}$.

## 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]

${\displaystyle y+z=z+y\qquad {\mbox{for all }}y,z\in \mathbb {R} }$
For example 4 + 5 = 5 + 4, since both expressions equal 9.
${\displaystyle yz=zy\qquad {\mbox{for all }}y,z\in \mathbb {R} }$
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, the logical biconditional function p ↔ q is equivalent to q ↔ p. This function is also written as p IFF q, or as p ≡ q, or as Epq.
The last form is an example of the most concise notation in the article on truth functions, which lists the sixteen possible binary truth functions of which eight are commutative: Vpq = Vqp; Apq (OR) = Aqp; Dpq (NAND) = Dqp; Epq (IFF) = Eqp; Jpq = Jqp; Kpq (AND) = Kqp; Xpq (NOR) = Xqp; Opq = Oqp.

### Noncommutative operations in everyday life

• Concatenation, the act of joining character strings together, is a noncommutative operation. For example
${\displaystyle EA+T=EAT\neq TEA=T+EA}$
• 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]

For example, the truth tables for f (A,B) = A Λ ¬B (A AND NOT B) and f (B,A) = B Λ ¬A are
A B f (A,B) f (B,A)
F F F F
F T F T
T F T F
T T F F
• Matrix multiplication is noncommutative since
${\displaystyle {\begin{bmatrix}0&2\\0&1\end{bmatrix}}={\begin{bmatrix}1&1\\0&1\end{bmatrix}}\cdot {\begin{bmatrix}0&1\\0&1\end{bmatrix}}\neq {\begin{bmatrix}0&1\\0&1\end{bmatrix}}\cdot {\begin{bmatrix}1&1\\0&1\end{bmatrix}}={\begin{bmatrix}0&1\\0&1\end{bmatrix}}}$

## Non-commuting operators in quantum mechanics

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

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

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 ${\displaystyle x}$ and ${\displaystyle -i\hbar {\frac {\partial }{\partial x}}}$ (where ${\displaystyle \hbar }$ is the reduced Planck constant). This is the same example except for the constant ${\displaystyle -i\hbar }$, so again the operators do not commute and the physical meaning is that the position and linear momentum in a given direction are complementary.

## Notes

1. Moore and Parker
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4. Axler, p.2
5. Gallian, p.34
6. p. 26,87
7. Krowne, p.1
8. Weisstein, Commute, p.1
9. Lumpkin, p.11
10. Gay and Shute, p.?
11. O'Conner and Robertson, Real Numbers
12. Cabillón and Miller, Commutative and Distributive
13. O'Conner and Robertson, Servois
14. Yark, p.1
15. Gallian p.236
16. Gallian p.250

## References

### Books

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

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

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Definition of commutativity and examples of commutative operations
• , Accessed 8 August 2007.
Explanation of the term commute
Examples proving some noncommutative operations
Article giving the history of the real numbers
Page covering the earliest uses of mathematical terms
Biography of Francois Servois, who first used the term