Difference between revisions of "Noncommutative"
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==Examples of noncommutative operations== | ==Examples of noncommutative operations== | ||
===Composition of functions=== | ===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 | + | 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>f(g(x))\ne 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 | + | For example, suppose <math>f(x) = x^2</math> and <math>g(x) = x+1</math>. Then <math>(f \circ g)(x)=f(g(x))=f(x+1)=(x+1)^2=x^2+2x+1</math>, and <math>(g \circ f)(x)=g(f(x))=g(x^2)=x^2+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>. |
===Matrix multiplication=== | ===Matrix multiplication=== |
Latest revision as of 15:30, 26 December 2017
Informally, noncommutative means "order matters".
More formally, if is some binary operation on a set, and
and
are elements of that set, then noncommutative means that
doesn't necessarily equal
.
Most common operations, such as addition and multiplication of numbers, are commutative. For example, , and
.
Contents
Examples of noncommutative operations
Composition of functions
If and
are functions, then usually,
. This can also be written
.
For example, suppose and
. Then
, and
. Unless
,
will not be the same as
.
Matrix multiplication
If and
are both
matrices, then usually,
. For example:
whereas
Symmetries of a regular n-gon
The symmetries of a regular n-gon form a noncommutative group called a dihedral group.