Binary Operations

These are operations involving two operands. Addition, subtraction, multiplication and division are all binary operations. You’re practically an expert at them!

Closure

A number set is closed under an operation if the result of any two elements of the set ALWAYS produces another element in the same number set.

Example: Real numbers under multiplication

Any real number multiplied by another real number will always produce another real number thus we say that the set of real numbers is closed under multiplication.

Example: Integers under division

We can easily pick any two integers e.g. $1$ and $2$ and show that the result of the division will NOT produce an integer: $$\frac12=0.5$$

Hence the set of integers is NOT closed under division.

Commutativity

A binary operation, $\oplus$ is commutative if the order of the operation between two elements does not matter: $$a\oplus b=b\oplus a$$

For example, multiplication is commutative: $$a\times b=b\times a$$

Subtraction is not: $$a-b\neq b-a$$

Associativity

A binary operation is associative if: $$(a\oplus b)\oplus c=a\oplus (b\oplus c)$$

Distributivity

One binary operation,$\otimes$ is distributive over another,$\odot$ if: $$a\otimes (b\odot c)=a\otimes b \odot a\otimes b$$

We know that multiplication is distributive over addition: $$a\times (b+ c)=a\times b + a\times b$$

Identity of an operation

The identity, $e$ of a binary operation, $\oplus$ is that single element which when operated on by every element $a$, produces that other element: $$a\oplus e=a$$

We already know of the additive identity, $0$: $$ \begin{equation}\begin{aligned} a+0=a\\ \end{aligned}\end{equation} $$ $0$ is the identity for addition because any number added to $0$ produces the number itself. We also know the multiplicative identity, $1$: $$ \begin{equation}\begin{aligned} a\times 1=a\\ \end{aligned}\end{equation} $$ $1$ is the identity for multiplication because any number times $1$ is the number itself.

Inverses

This is the element, $a^{-1}$ which when operated on by another element ($a$) produces the identity for that operation: $$ \begin{equation}\begin{aligned} a\oplus a^{-1}=e\\ \end{aligned}\end{equation} $$

Example: Addition

Because $0$ is the additive identity, it is easy to determine the inverse of any number, $a$ with respect to addition: $$ \begin{equation}\begin{aligned} a+a^{-1}&=e\\ a+a^{-1}&=0\\ a^{-1}&=0-a\\ \therefore a^{-1}&=-a\\ \end{aligned}\end{equation} $$

Thus for any number, $a$, the additive inverse of that number is simply the negative version of that number (a.k.a. the product of the number and $-1$).

Example: Multiplication

$1$ has already been established as the multiplicative identity thus the inverse ($a^{-1}$) can be found for any number, $a$: $$ \begin{equation}\begin{aligned} a\times a^{-1}&=e\\ a\times a^{-1}&=1\\ a^{-1}&=1\div a\\ \therefore a^{-1}&=\frac{1}{a}\\ \end{aligned}\end{equation} $$

For any number, $a$, the multiplicative inverse of that number is its reciprocal, written as $\frac{1}{a}$.