There are other axioms that can apply to binary operations.
Commutativity
A binary operation, $\oplus$ is commutative if the order of the operation between two elements does not matter: $$a\oplus b=b\oplus a$$
For example, multiplication is commutative: $$a\times b=b\times a$$
Subtraction is not: $$a-b\neq b-a$$
Associativity
A binary operation is associative if: $$(a\oplus b)\oplus c=a\oplus (b\oplus c)$$
Multiplication and addition are both known to be associative: $$ \begin{equation}\begin{aligned} (a+b)+c&=a+(b+c)\\ \therefore a+b+c&=a+b+c\\ \end{aligned}\end{equation} $$
$$ \begin{equation}\begin{aligned} (a\times b)\times c&=a\times (b\times c)\\ \therefore a\times b\times c&=a\times b\times c\\ \end{aligned}\end{equation} $$Independence of associativity and commutativity
It should be noted that if an operation is commutative, that does not mean it will be associative, and vice versa. Some operations that are associative but not commutative are:
- Matrix multiplication, $(AB)C=A(BC)$
- Composition of functions, $(f\circ g)\circ h=f\circ(g\circ h)$
Example
Determine if the operation $a*b=\frac{a+b}{2}$ is commutative and/or associative.
Solution Testing commutativity: $$ \begin{equation}\begin{aligned} a*b&=b*a\\ \frac{a+b}{2}&=\frac{b+a}{2}\\ \frac{a+b}{2}&=\frac{a+b}{2}\\ \end{aligned}\end{equation} $$
Testing associativity: $$ \begin{equation}\begin{aligned} (a*b)*c&=a*(b*c)\\ \frac{a+b}{2}*c&=a*\frac{b+c}{2}\\ \frac{\frac{a+b}{2}+c}{2}&=\frac{a+\frac{b+c}{2}}{2}\\ \frac{a+b}{4}+\frac{c}{2}&\neq\frac{a}{2}+\frac{b+c}{4}\\ \end{aligned}\end{equation} $$
Hence, the operation $*$ is commutative without being associative.
Distributivity
One binary operation,$\otimes$ is distributive over another,$\odot$ if: $$a\otimes (b\odot c)=a\otimes b \odot a\otimes b$$
We know that multiplication is distributive over addition: $$a\times (b+ c)=a\times b + a\times b$$
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