Binary Operations

A binary operation is an operation which combines two numbers to produce a third number. Our basic arithmetic operations are binary operations:

• Addition ($+$)
• Subtraction ($-$)
• Multiplication ($\times$)
• Division ($\div$)

Addition is a binary operation because it accepts two numbers and produces a third number (their sum) e.g. $2+1=3$.

Custom binary operations

We can define our own binary operations using fancy symbols like oplus ($\oplus$), otimes ($\otimes$), etc. For example, consider the binary operation:

$$ \begin{equation}\begin{aligned} a\oplus b = a+b-1\\ \end{aligned}\end{equation} $$

We can use this operation with pairs of inputs:

$$ \begin{equation}\begin{aligned} 5\oplus 3 &= 5+3-1\\ &=7\\ -4\oplus 2 &= -4+2-1\\ &=-3\\ \end{aligned}\end{equation} $$

At this level, using custom binary operations is simply a matter of substitution.

Challenge

Given the binary operation $\otimes$ defined as $x\otimes y=\frac{x-y}{2}$, evaluate the following:

• $1\otimes 2$
• $2\otimes 5$
• $3\otimes k$
• $x\otimes 0$
• $a^2\otimes b^3$

Closure property

A binary operation is said to be closed on a set if for every pair of elements in the set operating on each other, the result is an element of the same set. For example, consider the set of natural (a.k.a. counting) numbers ${1,2,3,…}$ with the operation of division ($\div$).

Division is only closed on the set of natural numbers if the result of every division is also a natural number.

By counter-example: $$ \begin{equation}\begin{aligned} 3\div 2 = 1.5\\ \end{aligned}\end{equation} $$

Here we divide a natural number $3$ by another natural number $2$. The result is $1.5$ which is NOT a natural number. This means that division is NOT closed on the set of natural numbers.

If we can pick any two numbers in the set that will not result in another number in the set then the operation is not closed on the set.

On the other hand, an operation like addition is closed on the set of natural numbers. This is because adding any two natural numbers will produce another natural number.

Challenge

Determine if the following operations are closed on the respective sets:

• $a\otimes b=a-b+1$ on the set of natural numbers, $\mathbb{N}$
• $a\oplus b=\sqrt{a}+b$ on the set of natural numbers, $\mathbb{N}$
• $x\ominus y=x\times (y-1)$ on the set of real numbers, $\mathbb{R}$

Hint: Try and find any pair of elements in the set that will not result in an element in the same set.