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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  can not be true for every pair of natural numbers  and .
For example  and .
2. Associative Operation: A binary operation a 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
and 
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
 for all 
and is said to be right distributive with respect to if
 for all 
Whenever the operation  is left as well as right distributive, we simply say that is distributive with respect .
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