What is a communitive property?
Commutative property - 2 reference results
Commutativity
In mathematics, commutativity is the ability to change the order
of something without changing the end result. It is a fundamental
property in most branches of mathematics and many proofs depend on
it. The commutativity of simple operations was for many years
implicitly assumed and the property was not given a name or
attributed until the 19th century when mathematicians began to
formalize the theory of mathematics. 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. In group and
set theory, many algebraic structures are called commutative when
certain operands satisfy the commutative property. In higher
branches of math, 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. The term "commutative" is used in
several related senses. 1. A binary operation ∗ on a set S is said
to be commutative if: : forall x,y in S: x * y = y * x , : - 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:A×A → B is said to be
commutative if: : forall x,y in A: f (x, y) = f(y, x) , 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 century 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. Simple
versions of the commutative property are usually taught in
beginning mathematics courses. The first use of the actual term
commutative was in a memoir by Francois 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 Philosophical Transactions of the Royal Society in 1844.
The associative property is closely related to the commutative
property. The associative property states that the order in which
operations are performed does not affect the final result. In
contrast, the commutative property states that the order of the
terms does not affect the final result. 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. *
Putting your shoes on resembles a commutative operation since it
doesn't matter if you put the left or right shoe on first, the end
result (having both shoes on), is the same. * When making change we
take advantage of the commutativity of addition. It doesn't matter
what order we put the change in, it always adds to the same total.
Two well-known examples of commutative binary operations are: * The
addition of real numbers, which is commutative since : y + z = z +
y quad forall y,zin mathbb{R} : For example 4 + 5 = 5 + 4, since
both expressions equal 9. * The multiplication of real numbers,
which is commutative since : y z = z y quad forall y,zin mathbb{R}
: For example, 3 × 5 = 5 × 3, since both expressions equal 15. *
Further examples of commutative binary operations include addition
and multiplication of complex numbers, addition of vectors, and
intersection and union of sets. * Washing and drying your clothes
resembles a noncommutative operation, if you dry first and then
wash, you get a significantly different result than if you wash
first and then dry. * The Rubik's Cube is noncommutative. For
example, twisting the front face clockwise, the top face clockwise
and the front face counterclockwise (FUF') does not yield the same
result as twisting the front face clockwise, then counterclockwise
and finally twisting the top clockwise (FF'U). The twists do not
commute. This is studied in group theory. Some noncommutative
binary operations are: * subtraction is noncommutative since 0-1neq
1-0 * division is noncommutative since 1/2neq 2/1 * 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} * An abelian group is a group whose
group operation is commutative. * A commutative ring is a ring
whose multiplication is commutative. (Addition in a ring is by
definition always commutative.) * In a field both addition and
multiplication are commutative.
Vivi
Maxine
Rafa
