Binary Operations

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 \mathbb{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 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 Gwhere the image of \left(  {a,b} \right) of G \times G under “ * ”, i.e.,  * \left( {a,b} \right), is denoted by a * b.

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 \mathbb{N}, because if m,n \in \mathbb{N} then m + n \in \mathbb{N} 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 ifm,n \in \mathbb{N} then m - n may not belongs 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 (-) both are 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 arc 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.