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 m,n another natural number m+n, similarly the multiplication operation gives for the pair m,n the number m,n in N 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 G is a rule which associates to each pair of elements a,b in G a unique element 
of G.Alternatively a binary operation “
” on G is a mapping from 
to G i.e. 
where the image of (a,b) 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.
on a non-empty set G is a rule which associates to each pair of elements a,b in G a unique element of G.Alternatively a binary operation “” on G is a mapping from to G i.e. where the image of (a,b) 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 G, we say that G is closed or closure property is satisfied in G, with respect to the operation “
”.
(1) Usual addition (+) is binary operation on N, 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 m-n may not belongs to N. For example if 
and 
their m-n=5-6=-1 which does not belong to N. (2) Usual addition ( ) and usual subtraction (-) both are binary operations on Z because if
then 
and
. (3) Union, intersection and difference arc binary operations on P(A), the power set of A.
(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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