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 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. $$ * :G \times G \to G$$ 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 “$$ * $$”.

 

Examples:

(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.