# Types of Binary Operations

1. Commutative Operation:

A binary operation $*$ over a set $G$ is said to be commutative, if for every pair of elements $a,b \in G$, $a * b = b * a$

Thus addition and multiplication are commutative binary operations for natural numbers whereas subtraction and division are not commutative because, for $a - b = b - a$ and $a \div b = b \div a$ cannot be true for every pair of natural numbers $a$ and $b$.

For example $5 - 4 \ne 4 - 5$ and$5 \div 4 \ne 4 \div 5$.

2. Associative Operation:

A binary operation a on a set $G$ is called associative if $a * \left( {b * c} \right) = \left( {a * b} \right) * c$ for all $a,b,c \in G$.

Evidently ordinary addition and multiplication are associative binary operations on the set of natural numbers, integers, rational numbers and real numbers. However, if we define $a * b = a - 2b{\text{ }}\forall a,b \in \mathbb{R}$, then

$\left( {a * b} \right) * c = \left( {a * b} \right) - 2c = \left( {a - 2b} \right) - 2c = a - 2b - 2c$

And

$a * \left( {b * c} \right) = a - 2\left( {b * c} \right) = a - 2\left( {b - 2c} \right) = a - 2b - 2c$

Thus, the operation defined as above is not associative.

3. Distributive Operation:

Let $*$ and $* '$ be two binary operations defined on a set $G$. Then the operation $* '$ is said to be left distributive with respect to operation $*$ if

$a * '\left( {b * c} \right) = \left( {a * 'b} \right) * \left( {a * 'c} \right)$ for all $a,b,c \in G$

and is said to be right distributive with respect to $*$ if

$\left( {b * c} \right) * 'a = \left( {b * 'a} \right) * \left( {c * 'a} \right)$ for all $a,b,c \in G$

Whenever the operation $* '$ is left as well as right distributive, we simply say that $* '$ is distributive with respect $*$.