Definitions of Binary OperationDef1:It is the generalization of the standard operation like a addition and multiplication on a set of numbers.

Def2: It is a function G*GG Def3: Let G be a set and * denote a binary operation. So a given

a,b G, we have an unique a*b G.

Cont.

Def4: A cartesian product of the set A and B, where neither

A or B is the empty set, is a set of order pair(x,y) such that

x A and y B (x,y)

Examples:

1- {a,b,c} *{b,d}= {(a,b),(a,d),(b,b),(b,d),(c,b),(c,d)}

2- If S= {1,5}, S*S={(1,1),(1,5),(5,1),(5,5)}

Properties of Binary OperationClousure: It is said to be closed on a set G iff x A and y A then x *y is also an element of A.Commutivity: It is said to be commutative iff

x* y = y*x , for every x and y in G.

Cont.Associativity: It is said to be associative on the set G iff

(x*y)*z= x*(y*z), for every x,y,z in G.

Identity: It is said to possess an identity iff there exists an element e in G such as:

x*e= x and e*x= x, for every x in G

Cont.Inverse: An element x in G is said to possess an inverse( which will be denoted x-1) iff

a) x* x-1= e b) x-1*x= e

Cont.

Distributivity: It uses two binary operations on the same set.If the operation * and are defined on the set G such that:

a*(b c)= (a*b ) (a*c), for all a,b,c G