Binary Operations

We will cover binary operations under this 5 aspects- Definition, Commutativity, Associativity, Identity and Inverse.
1)Definition- A binary operation or dyadic operation is a calculation that combines two elements (called operands) to produce another element. More specifically, a binary operation on a set is a binary operation whose two domains and the codomain are the same set.
2) Commutativity- In mathematics, a binary operation is commutative if changing the order of the operations does not change the result. That is a*b= b*a ∀a, b∈A, here '*' means a binary operation on A.
3) Associativity- It means if take 3 elements at a time then in which order we proceed doesn't matter. This means (a*b)*c= a*(b*c) ∀ a,b,c ∈ A.
4) Identity- This aspect says that anything like addition with indentity or multiplication with indentity will gave you same orignal number. ∃e∈ A such that a*e= e*a =a ∀ a,b,c ∈ A.
5) Inverse- For a given element in the set, you would say there exists b belonging to the same set such that a*b= b*a and it gives us the orignal identity element. This means for a∈ A,            ∃ b∈ A such that a*b= b*a= e.
Example- Let `*' is a binary operation on set of all non- zero real numbers, given by       

`a"*"b= (ab)/5` ∀ a,b,c ∈ R-{0}. Find x, given 2* (x*5)= 10

Solution- `2"*" (5x)/5= 10`

`(2x)/5= 10`

`x=25`