Difference between revisions of "Noncommutative"

Revision as of 16:52, 25 April 2008

Informally, noncommutative means "order matters".

More formally, if $\star$ is some binary operation on a set, and $x$ and $y$ are elements of that set, then noncommutative means that $x \star y$ doesn't necessarily equal $y \star x$ .

Most common operations, such as addition and multiplication of numbers, are commutative. For example, $4\cdot3=3\cdot4=12$ , and $2+3=3+2=5$ .

Examples of noncommutative operations

Composition of functions

If $f(x)$ and $g(x)$ are functions, then usually, $(f\circ g)(x)\ne(g\circ f)(x)$ . This can also be written $g(f(x))\ne f(g(x))$ .

For example, suppose $f(x) = x^2$ and $g(x) = x+1$ . Then $(f\circ g)(x)=g(f(x))=g(x^2)=x^2+1$ , and $(g\circ f)(x)=f(g(x))=f(x+1)=(x+1)^2=x^2+2x+1$ . Unless $x=0$ , $(f\circ g)(x)$ will not be the same as $(g\circ f)(x)$ .

Matrix multiplication

If $A$ and $B$ are both $n\times n$ matrices, then usually, $AB\ne BA$ . For example:

$\begin{pmatrix}1&2\\3&4\end{pmatrix}\begin{pmatrix}5&6\\7&8\end{pmatrix}=\begin{pmatrix}19&22\\43&50\end{pmatrix}$

whereas

$\begin{pmatrix}5&6\\7&8\end{pmatrix}\begin{pmatrix}1&2\\3&4\end{pmatrix}=\begin{pmatrix}23&34\\31&46\end{pmatrix}$

Symmetries of a regular n-gon

The symmetries of a regular n-gon form a noncommutative group called a dihedral group.

@@ Line 1: / Line 1: @@
-Informally, '''noncommutative''' means that "order matters".
+Informally, '''noncommutative''' means "order matters".
-More formally, if <math>\displaystyle\star</math> is some [[binary operation]] on a  [[set]], and x and y are elements of the set, then noncommutative means that <math>\displaystyle x \star y</math> doesn't necessarily equal <math>\displaystyle y \star x</math>.
+More formally, if <math>\star</math> is some [[binary operation]] on a [[set]], and <math>x</math> and <math>y</math> are elements of that set, then noncommutative means that <math>x \star y</math> doesn't necessarily equal <math>y \star x</math>.
-Most common operations, such as addition or multiplication of numbers, are of course [[commutative]]:  for example, 3+2 = 2+3 and 6x8 = 8x6.
+Most common operations, such as [[addition]] and [[multiplication]] of numbers, are [[commutative]]. For example, <math>4\cdot3=3\cdot4=12</math>, and <math>2+3=3+2=5</math>.
 ==Examples of noncommutative operations==
 ===Composition of functions===
+If <math>f(x)</math> and <math>g(x)</math> are functions, then usually, <math>(f\circ g)(x)\ne(g\circ f)(x)</math>. This can also be written <math>g(f(x))\ne f(g(x))</math>.
-If <math>f(x)</math> and <math>g(x)</math> are functions, then usually <math>(f \circ g)(x) \not= (g \circ f)(x)</math>, or to write it another way, <math>f(g(x)) \not= g(f(x))</math>.
+For example, suppose <math>f(x) = x^2</math> and <math>g(x) = x+1</math>.  Then <math>(f\circ g)(x)=g(f(x))=g(x^2)=x^2+1</math>, and <math>(g\circ f)(x)=f(g(x))=f(x+1)=(x+1)^2=x^2+2x+1</math>. Unless <math>x=0</math>, <math>(f\circ g)(x)</math> will not be the same as <math>(g\circ f)(x)</math>.
-For example, suppose <math>f(x) = x^2</math> and <math>g(x) = x+1</math>.  Then <math>(f \circ g)(x) = g(f(x)) = g(x^2) = x^2 + 1</math>, whereas <math>(g \circ f)(x) = f(g(x)) = f(x+1) = (x+1)^2 = x^2+2x+1</math>, and these are clearly not equal!
 ===Matrix multiplication===
+If <math>A</math> and <math>B</math> are both <math>n\times n</math> [[matrix|matrices]], then usually, <math>AB\ne BA</math>.  For example:
-If A and B are both n-by-n [[matrix|matrices]], then usually <math>AB \not= BA</math>.  For example:
+<math>\begin{pmatrix}1&2\\3&4\end{pmatrix}\begin{pmatrix}5&6\\7&8\end{pmatrix}=\begin{pmatrix}19&22\\43&50\end{pmatrix}</math>
-<math>
-\begin{pmatrix}1&2\\3&4\end{pmatrix}
-\begin{pmatrix}5&6\\7&8\end{pmatrix} =
-\begin{pmatrix}19&22\\43&50\end{pmatrix}
-</math>
 whereas
-<math>
+<math>\begin{pmatrix}5&6\\7&8\end{pmatrix}\begin{pmatrix}1&2\\3&4\end{pmatrix}=\begin{pmatrix}23&34\\31&46\end{pmatrix}</math>
-\begin{pmatrix}5&6\\7&8\end{pmatrix}
-\begin{pmatrix}1&2\\3&4\end{pmatrix} =
-\begin{pmatrix}23&34\\31&46\end{pmatrix}
-</math>
 ===Symmetries of a regular n-gon===
+The symmetries of a regular n-gon form a noncommutative [[group]] called a [[dihedral group]].
-The symmetries of a regular n-gon form a noncommutative [[group]] called a [[dihedral group]].
+[[Category:Definition]]

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