Commutative property explained

Commutative property
Type:Property
Field:Algebra
Statement:A binary operation is commutative if changing the order of the operands does not change the result.
Symbolic Statement:

x*y=y*x   \forallx,y\inS.

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. Perhaps most familiar as a property of arithmetic, e.g. or, 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,); 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 centuries implicitly assumed. Thus, this property was not named until the 19th century, when new algebraic structures started to be studied.

Definition

*

on a set S is commutative ifx * y = y * x for all

x,y\inS

. An operation that is not commutative is said to be noncommutative.

One says that commutes with or that and commute under

*

if x * y = y * x.

So, an operation is commutative if every two elements commute. An operation is noncommutative if there are two elements such that

x*y\ney*x.

This does not exclude the possibility that some pairs of elements commute.

Examples

Commutative operations

Noncommutative operations

1 ÷ 2 ≠ 2 ÷ 1

. Subtraction is noncommutative, since

0-1 ≠ 1-0

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

x-y=-(y-x)

for every and . Exponentiation is noncommutative, since

23 ≠ 32

(see Equation xy = yxEquation xy = yx).

f(x)=2x+1

and

g(x)=3x+7

. Then (f \circ g)(x) = f(g(x)) = 2(3x+7)+1 = 6x+15 and (g \circ f)(x) = g(f(x)) = 3(2x+1)+7 = 6x+10.

\begin 0 & 2 \\ 0 & 1 \end = \begin 1 & 1 \\ 0 & 1 \end \begin 0 & 1 \\ 0 & 1 \end \neq \begin 0 & 1 \\ 0 & 1 \end \begin 1 & 1 \\ 0 & 1 \end = \begin 0 & 1 \\ 0 & 1 \end

b x a=-(a x b)

.

Commutative structures

Some types of algebraic structures involve an operation that does not require commutativity. If this operation is commutative for a specific structure, the structure is often said to be commutative. So,

However, in the case of algebras, the phrase "commutative algebra" refers only to associative algebras that have a commutative multiplication.

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. Euclid is known to have assumed the commutative property of multiplication in his book Elements.[1] 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. Nowadays, 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. Commutative is the feminine form of the French adjective commutatif, which is derived from the French noun commutation and the French verb commuter, meaning "to exchange" or "to switch", a cognate of to commute. The term then appeared in English in 1838. in Duncan Gregory's article entitled "On the real nature of symbolical algebra" published in 1840 in the Transactions of the Royal Society of Edinburgh.

See also

References

Notes and References

  1. . See Book VII, Proposition 5, in David E. Joyce's online edition of Euclid's Elements

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