The concept of binary operations 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 an ally two natural numbers another natural number . Similarly, multiplication gives for the pair the number in again. These types of operations are 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 .
Simply put, 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 the closure property is satisfied in with respect to the operation “”.
(1) Usual addition (+) is a binary operation on , because if then as we know that the sum of two natural numbers is again a natural number. But the usual subtraction (-) is not a binary operation on N because if then may not belong to . For example, if and their , which does not belong to .
(2) Usual addition (+) and usual subtraction (-) are both binary operations on , because if then and .
(3) Union, intersection and difference are 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.