Chapter 9, Problem 1RE

What is a binary operation?

To determine

To define: The binary operation.

Explanation of Solution

A binary operation is an operation, or rule, that can be performed on two and only two elements (input) of the set. The resulting element (output) is a single element.

Add two elements, the resulting sum is a single element. Likewise for multiplication.

Examples for addition is $1+5=6$.

Example for multiplication is $1\times 5=5$.

But subtraction and division only for exceptional case.

Example for subtraction $5-8$ is not binary operation in natural numbers $\mathbb{N}$ because $5-8=-3$ not in $\mathbb{N}$.

Division is a binary operation only on $\mathbb{Q}\backslash \left\{0\right\}$. Since 0 is undefined.

Note:

A reciprocal of a number is not a binary operation, since it is an operation on a single element.