Noncommutative

From AoPSWiki

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 $(g\circ f)(x)$ .