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 and5 \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  * .   

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