Information AboutCommutative |
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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'' . For example: : for ''all'' Real Number s x and y. Otherwise, the operation is noncommutative: 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 . The most well-known examples of commutative binary operations are Addition and Multiplication of Real Number s; for example:
Further examples of commutative binary operations include addition and multiplication of Complex Number s, addition of Vectors , and Intersection and Union of Set s. In each case, these operations are commutative over their entire domains. Among the noncommutative binary operations are Subtraction (''a'' − ''b''), Division (''a''/''b''), Exponentiation (''a''''b''), Function Composition (''f'' o ''g''), Tetration (''a''↑↑''b''), Matrix multiplication, and Quaternion multiplication. A real life example of noncommutativity is the . The subset of the domain on which an operation is commutative is sometimes called the Center in algebra. 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. 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 . 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 ). NEUROPHYSIOLOGICAL MEANING In Neurophysiology , ''commutative'' has much the same meaning as in algebra. Physiologist Douglas A. Tweed and coworkers consider whether certain neural circuits in the Brain exhibit noncommutativity and state: :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 Circuit s that steer the Eye s, 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. SEE ALSO |
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