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The concept of binary operation on a set is a generalization of the standard operations like addition and multiplication on the set of numbers. For instance we know that the operation of addition (+) gives for ally two natural numbers  another natural number , similarly the multiplication operation gives for the pair  the number  in  again. These types of operations arc found to exist in many other sets. Thus we give the following definition.
Binary Operation:
A binary operation to be denoted by  on a non-empty set  is a rule which associates to each pair of elements  in  a unique element  of  .
Alternatively a binary operation “ ” on  is a mapping from  to i.e.  where the image of  of  under “ ”, i.e.,  , is denoted by  .
Thus in simple language we may say that a binary operation on a set tells us how to combine any two elements of the set to get a unique element, again of the same set.
If an operation “ ” is binary on a set , we say that is closed or closure property is satisfied in , with respect to the operation “ ”.
Examples:
(1) Usual addition (+) is binary operation on , because if  then  as we know that sum of two natural numbers is again a natural number. But the usual subtraction (-) is not binary operation on N because if then may not belongs to . For example if  and  their  which does not belong to .
(2) Usual addition (+) and usual subtraction (-) both are binary operations on  because if  then  and .
(3) Union, intersection and difference arc binary operations on , the power set of .
(4) Vector product is a binary operation on the Set of all 3-Dimensional Vectors but the dot product is not a binary operation as the dot product is not a vector but a scalar.
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