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Mathematical meaning
A map or binary operation  is said to be commutative when, for any x in A and any y in B
f(x,y) = f(y,x). In mathematics and related technical fields, the term map or mapping is often a synonym for function. ...
For example:
 for all real numbes x and y.
Otherwise, the operation is noncommutative:
x − y = y − x
2x − 2y = 0
2(x − y) = 0
x − y = 0
x = y
So, subtraction is commutative if and only if x = y and noncommutative for any other pair of real numbers.
Additionally, if
for a particular pair of elements x and y, then x and y are said to commute. Every element commutes with itself and, in a group, every element commutes with the identity, with its own inverse, and with its powers. In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
// Computer programming In object-oriented programming, object identity is a mechanism for distinguishing different objects from each other. ...
Inverse typically means the opposite of something. ...
In mathematics, exponentiation (frequently known colloquially as raising a number to a power) is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition. ...
The most well-known examples of commutative binary operations are addition and multiplication of real numbers; for example: 3 + 2 with apples, a popular choice in textbooks Addition is the basic operation of arithmetic. ...
In mathematics, multiplication is an arithmetic operation which is the inverse of division, and in elementary arithmetic, can be interpreted as repeated addition. ...
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
- 4 + 5 = 5 + 4 (since both expressions evaluate to 9)
- 2 × 3 = 3 × 2 (since both expressions evaluate to 6)
Further examples of commutative binary operations include addition and multiplication of complex numbers, addition of vectors, and intersection and union of sets. In each case, these operations are commutative over their entire domains. An expression in the very basic sense is the noun form of the verb express. ...
Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ...
Vector spaces (or linear spaces) are spaces whose elements, known as vectors, can be scaled and added; all linear combinations can be formed. ...
In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ...
In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...
In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...
Among the noncommutative binary operations are subtraction (a − b), division (a/b), exponentiation (ab), function composition (f o g), tetration (a↑↑b), matrix multiplication, and quaternion multiplication. 5 - 2 = 3 Subtraction is one of the four basic arithmetic operations; it is essentially the opposite of addition. ...
In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication, and sometimes it can be interpreted as repeated subtraction. ...
In mathematics, exponentiation (frequently known colloquially as raising a number to a power) is a process generalized from repeated (or iterated) multiplication, in much the same way that multiplication is a process generalized from repeated addition. ...
In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...
Tetration (also exponential map, hyperpower, power tower, super-exponentiation, and hyper4) is iterated exponentiation, the first hyper operator after exponentiation. ...
In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, of elements of a ring-like algebraic structure. ...
In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...
A real life example of noncommutativity is the Rubik's Cube: 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 don't commute. This is studied in group theory. Rubiks Cube in scrambled state. ...
Group theory is that branch of mathematics concerned with the study of groups. ...
The subset of the domain on which an operation is commutative is sometimes called the center in algebra. The term center is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. ...
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 always commutative.) In a field both addition and multiplication are commutative. In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ...
In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...
In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ...
In mathematics, multiplication is an arithmetic operation which is the inverse of division, and in elementary arithmetic, can be interpreted as repeated addition. ...
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
Commutativity can be another name for symmetry. That is, suppose we solve a problem involving parameters x and y, and determine that the solution is equal to f(x,y). If there exists a subset of values for x and y where the two values can be exchanged without affecting the function, the problem is symmetric. Many symmetries arise naturally in mathematics out of simpler symmetries, and are commonly found useful for particular kinds of proofs (see WLOG). Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ...
Without loss of generality or simply WLOG is a frequently used expression in mathematics. ...
Neurophysiological meaning In neurophysiology, commutative has much the same meaning as in algebra. Neurophysiology is a part of physiology as a science, which is concerned with the study of the nervous system. ...
Physiologist Douglas A. Tweed and coworkers consider whether certain neural circuits in the brain exhibit noncommutativity and state: Comparative brain sizes In animals, the brain, or encephalon (Greek for in the head), is the control center of the central nervous system. ...
- In non-commutative algebra, order makes a difference to multiplication, so that
 . This feature is necessary for computing rotary motion, because order makes a difference to the combined effect of two rotations. It has therefore been proposed that there are non-commutative operators in the brain circuits that deal with rotations, including motor circuits that steer the eyes, head and limbs, and sensory circuits that handle spatial information. This idea is controversial: studies of eye and head control have revealed behaviours that are consistent with non-commutativity in the brain, but none that clearly rules out all commutative models. (Douglas A. Tweed and others, Nature 399, 261 - 263; 20 May 1999). Tweed goes on to demonstrate non-commutative computation in the vestibulo-ocular reflex by showing that subjects rotated in darkness can hold their gaze points stable in space---correctly computing different final eye-position commands when put through the same two rotations in different orders, in a way that is unattainable by any commutative system. Rotation of a plane, seen as the rotation of the terrain relative to the plane (exposure time 1. ...
This article or section does not cite its references or sources. ...
For other uses of the word head, see head (disambiguation). ...
Senses are the physiological methods of perception. ...
First title page, November 4, 1869 Nature is one of the oldest and most reputable scientific journals, first published on 4 November 1869. ...
20 May is the 140th day of the year in the Gregorian Calendar (141st in leap years). ...
1999 (MCMXCIX) was a common year starting on Friday, and was designated the International Year of Older Persons by the United Nations. ...
Figure 3 Three-neuron arc, during a head movement to the right. ...
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