# 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 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 to $$ * $$.