Commutative property - Revision history

GregG: fix a punctuation error

2013-08-06T02:24:57Z

fix a punctuation error

GregG: add an example of how commutative and associativity are distinct

2012-01-28T02:02:47Z

add an example of how commutative and associativity are distinct

DavidE: internal links

2012-01-27T20:58:28Z

internal links

DavidE: discussed anticommutative operations

2012-01-27T20:56:59Z

discussed anticommutative operations

DavidE: applicability to vectors

2012-01-25T15:01:53Z

applicability to vectors

Hsmom: bits and pieces

2011-04-22T02:39:28Z

bits and pieces

LanceS: Undo revision 827324 by Karajou (Talk) this is correct: e.g. complex numbers with the operation (a,b) -> conj(ab) is commutative but not assoc

2010-11-09T06:02:55Z

Undo revision 827324 by Karajou (Talk) this is correct: e.g. complex numbers with the operation (a,b) -> conj(ab) is commutative but not assoc

Karajou: Reverted edits by ELWisty (Talk) to last revision by CherryS

2010-11-09T05:30:31Z

Reverted edits by ELWisty (Talk) to last revision by CherryS

ELWisty: Corrected one grammatical and one factual error.

2010-08-30T00:00:26Z

Corrected one grammatical and one factual error.

CherryS at 21:26, November 23, 2008

2008-11-23T21:26:13Z

@@ Line 3: / Line 3: @@
 In layman's terms, an equation demonstrates commutativity when the constants or variables can be moved around an operation without changing the answer (e.g. 1 + 2 = 2 + 1  or  2 * 3 = 3 * 2). It is as if the numbers are "commuting" from one place to another.  Thus, using real numbers, both multiplication and addition are commutative, and subtraction and division are not.  For [[vector]]s, the [[dot product]] is commutative, while the [[cross product]] is anticommutative.
-The commutative property does not imply the [[associative property]], nor vice versa.  For example, matrix multiplication is associative but not commutative.  Also, define the function <math>\Box</math> on integers by <math>a \Box b = 2(a + b)</math>.  Although, <math>\Box</math> is commutative, it is not associative (as <math>((1 \Box 0) \Box 0) = 2 \Box 0 = 4</math> but <math>(1 \Box (0 \Box 0)) = 1 \Box 0 = 2</math>.
+The commutative property does not imply the [[associative property]], nor vice versa.  For example, matrix multiplication is associative but not commutative.  Also, define the function <math>\Box</math> on integers by <math>a \Box b = 2(a + b)</math>.  Although, <math>\Box</math> is commutative, it is not associative, as <math>((1 \Box 0) \Box 0) = 2 \Box 0 = 4</math> but <math>(1 \Box (0 \Box 0)) = 1 \Box 0 = 2</math>.
 [[Category:Mathematics]]

@@ Line 3: / Line 3: @@
 In layman's terms, an equation demonstrates commutativity when the constants or variables can be moved around an operation without changing the answer (e.g. 1 + 2 = 2 + 1  or  2 * 3 = 3 * 2). It is as if the numbers are "commuting" from one place to another.  Thus, using real numbers, both multiplication and addition are commutative, and subtraction and division are not.  For [[vector]]s, the [[dot product]] is commutative, while the [[cross product]] is anticommutative.
-The commutative property <strike>implies</strike> does not imply the [[associative property]].
+The commutative property does not imply the [[associative property]], nor vice versa.  For example, matrix multiplication is associative but not commutative.  Also, define the function <math>\Box</math> on integers by <math>a \Box b = 2(a + b)</math>.  Although, <math>\Box</math> is commutative, it is not associative (as <math>((1 \Box 0) \Box 0) = 2 \Box 0 = 4</math> but <math>(1 \Box (0 \Box 0)) = 1 \Box 0 = 2</math>.
 [[Category:Mathematics]]

@@ Line 1: / Line 1: @@
 In mathematics, the '''commutative property''' states that a [[binary operation]] <math>*</math> on a set '''A''' is said to be commutative if for all <math>x,y</math> in '''A'''  we have <math>x*y=y*x</math>. An example of a commutative operation is [[addition]] in the set of [[real numbers]]. When a [[Group (mathematics)|group]]'s operation is commutative, it is said to be [[abelian]].  If <math>x*y=-(y*x)</math>, then the operation is said to be '''anticommutative'''.
-In layman's terms, an equation demonstrates commutativity when the constants or variables can be moved around an operation without changing the answer (e.g. 1 + 2 = 2 + 1  or  2 * 3 = 3 * 2). It is as if the numbers are "commuting" from one place to another.  Thus, using real numbers, both multiplication and addition are commutative, and subtraction and division are not.  For [[vector]]s, the dot product is commutative, while the cross product is anticommutative.
+In layman's terms, an equation demonstrates commutativity when the constants or variables can be moved around an operation without changing the answer (e.g. 1 + 2 = 2 + 1  or  2 * 3 = 3 * 2). It is as if the numbers are "commuting" from one place to another.  Thus, using real numbers, both multiplication and addition are commutative, and subtraction and division are not.  For [[vector]]s, the [[dot product]] is commutative, while the [[cross product]] is anticommutative.
 The commutative property <strike>implies</strike> does not imply the [[associative property]].
 [[Category:Mathematics]]

@@ Line 1: / Line 1: @@
-In mathematics, the '''commutative property''' states that a [[binary operation]] <math>*</math> on a set '''A''' is said to be commutative if for all <math>x,y</math> in '''A'''  we have <math>x*y=y*x</math>. An example of a commutative operation is [[addition]] in the set of [[real numbers]]. When a [[Group (mathematics)|group]]'s operation is commutative, it is said to be [[abelian]].
+In mathematics, the '''commutative property''' states that a [[binary operation]] <math>*</math> on a set '''A''' is said to be commutative if for all <math>x,y</math> in '''A'''  we have <math>x*y=y*x</math>. An example of a commutative operation is [[addition]] in the set of [[real numbers]]. When a [[Group (mathematics)|group]]'s operation is commutative, it is said to be [[abelian]].  If <math>x*y=-(y*x)</math>, then the operation is said to be '''anticommutative'''.
-In layman's terms, an equation demonstrates commutativity when the constants or variables can be moved around an operation without changing the answer (e.g. 1 + 2 = 2 + 1  or  2 * 3 = 3 * 2). It is as if the numbers are "commuting" from one place to another.  Thus, using real numbers, both multiplication and addition are commutative, and subtraction and division are not.  For [[vector]]s, the dot product is commutative, while the cross product is not.
+In layman's terms, an equation demonstrates commutativity when the constants or variables can be moved around an operation without changing the answer (e.g. 1 + 2 = 2 + 1  or  2 * 3 = 3 * 2). It is as if the numbers are "commuting" from one place to another.  Thus, using real numbers, both multiplication and addition are commutative, and subtraction and division are not.  For [[vector]]s, the dot product is commutative, while the cross product is anticommutative.
 The commutative property <strike>implies</strike> does not imply the [[associative property]].
 [[Category:Mathematics]]

@@ Line 1: / Line 1: @@
 In mathematics, the '''commutative property''' states that a [[binary operation]] <math>*</math> on a set '''A''' is said to be commutative if for all <math>x,y</math> in '''A'''  we have <math>x*y=y*x</math>. An example of a commutative operation is [[addition]] in the set of [[real numbers]]. When a [[Group (mathematics)|group]]'s operation is commutative, it is said to be [[abelian]].
-In layman's terms, an equation demonstrates commutativity when the constants or variables can be moved around an operation without changing the answer (e.g. 1 + 2 = 2 + 1  or  2 * 3 = 3 * 2). It is as if the numbers are "commuting" from one place to another.  Thus, using real numbers, both multiplication and addition are commutative, and subtraction and division are not.
+In layman's terms, an equation demonstrates commutativity when the constants or variables can be moved around an operation without changing the answer (e.g. 1 + 2 = 2 + 1  or  2 * 3 = 3 * 2). It is as if the numbers are "commuting" from one place to another.  Thus, using real numbers, both multiplication and addition are commutative, and subtraction and division are not.  For [[vector]]s, the dot product is commutative, while the cross product is not.
 The commutative property <strike>implies</strike> does not imply the [[associative property]].
 [[Category:Mathematics]]