# 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 “$*$”.

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.