1. Commutative Operation:
A binary operation over a set is said to be commutative if for every pair of elements ,
Thus addition and multiplication are commutative binary operations for natural numbers whereas subtraction and division are not commutative, because for and cannot be true for every pair of natural numbers and .
For example and.
2. Associative Operation:
A binary operation on a set is called associative if for all .
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 , then
Thus, the operation defined as above is not associative.
3. Distributive Operation:
Let and be two binary operations defined on a set . Then the operation is said to be left distributive with respect to operation if
and is said to be right distributive with respect to if
Whenever the operation is left as well as right distributive, we simply say that is distributive with respect to .