Binary Operations

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 m,n another natural number m + n. Similarly, multiplication gives for the pair m,n the number m.n in \mathbb{N} again. These types of operations are found to exist in many other sets. Thus we give the following definition.


Binary OperationA 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 a * b of G.

Alternatively a binary operation “ * ” on G is a mapping from G \times G to G i.e.  * :G \times G \to G where the image of \left( {a,b} \right) of G \times G under “ * ”, i.e.,  * \left( {a,b} \right), is denoted by a * b.

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 G, we say that G is closed or the closure property is satisfied in G with respect to the operation “ * ”.



(1) Usual addition (+) is a binary operation on \mathbb{N}, because if m,n \in \mathbb{N} then m + n \in \mathbb{N} 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 m,n \in \mathbb{N} then m - n may not belong to \mathbb{N}. For example, if m = 5 and n = 6 their m - n = 5 - 6 = - 1, which does not belong to \mathbb{N}.
(2) Usual addition (+) and usual subtraction (-) are both binary operations on \mathbb{Z}, because if m,n \in \mathbb{Z} then m + n \in \mathbb{Z} and m - n \in \mathbb{Z}.
(3) Union, intersection and difference are binary operations on P\left( A \right), 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.