Types of Binary Operations With Example Updated 2021

Let's talk about types of Binary Operations

The First one is

Commutative Operation:

A binary operation is Commutative if, For every pair of element a,b ∊ M,

So

a o b = b o a

Addition and multiplications for natural numbers are commutative binary operations. But subtraction and division are not

Because for a, b ∊ a,
and a - b ≠ b - a
and a ÷ b ≠ b ÷ a

Here is a example

3-2 ≠ 2-3
3 ÷ 2 ≠ 2 ÷ 3

Associative Operation:

In associative operation is a calculation from which we get the same result regardless of the way the numbers are grouped. Addition and multiplication are both associative, while subtraction and division are not.

Here is a example:

A binary operation is associative if o on set M is like

a o (b o c) = (a o b) o c

for all

a, b, c ∊ M

Distributive Operation:

Ok so Binary operations are distributive if

// Consider * as multiplication and o as subtraction.

a * ( b o c ) = ( a * b ) o ( a * c )

or

( b o c ) * a = ( b * a ) o ( c * a )

So

a = 2, b = 6, c = 5
then 
a * ( b o c ) = a x (b - c ) = 2 x ( 6 - 5 ) = 2

Identity Operation

An identity operation is a operation when The element which, when combined with another element using an operation, leaves the element unchanged.

Here an example:

y + 0 = 0 + y = y

Inverse Element

An element a in a set with a binary operation, an inverse element for a is an element which gives the identity when composed with. a .

example:

b + (– b) = 0 = (– b) + b, 

–b is the inverse of b for addition