Binary operation: Definition and their laws

"A Binary operation or Binary composition 'o' for a set 'A' is a rule by which every ordered pair (a,b) is associated with a uniquely determined element of 'A', where a ᗴ A and b ᗴ A."

Thus a Binary operation 'O' on the set 'A' is a mapping from the set A X A to the set A

 i.e. O : A X A ➜ A.

Example :

 1. Ordinary Addition (+)  : Ordinary Addition for real number (R) is a binary operation, because 
(a + b) ᗴ  R ; where a ᗴ R and b ᗴ R.

2. Ordinary Subtraction (-) : Ordinary Subtraction for real number is a binary operation over real number (R); but it is not a binary operation over natural number (N), because 
2 ᗴ N , 3 ᗴ N  but ( 2-3 ) ∉ N.

3. Ordinary Multiplication(*) :Ordinary Multiplication for integers (Z) is a binary operation over the set of integers, but it is not a binary operation over the set of negative integers, because the product of two negative integers is not a negative integer.

4. Ordinary Division(/) : Ordinary Division for real numbers is a binary operation over R - {0}, but not a binary operation over the set of integers (Z).



Laws or Properties of Binary Operations

 1. Clouser Law : Let 'A' be a non-empty set on binary operation 'O' is defined, then we say that the set 'A' is cloused under the operation 'O'. Thus the set 'A' is coused under the binary operation if
a O b ᗴ A ∀ a, b ᗴ A.

Example : The set N of natural number is cloused under the binary operation '+' (ordinary addition) , but N is not cloused under the binary operation '-' (ordinary subtraction), because
4 ᗴ N, 5 ᗴ N doesn't implies (4-5) ᗴ N.

2. Commutative Law : Let 'A' be a non-empty set on which a binary operation 'O' is defined such that; ( a O b ) = ( b O a ) ∀ a, b ᗴ A.

Example : Let R be the set of non-empty real numbers, then '*' (ordinary multiplication) is binary operation over R,
      also ( a * b )  = ( b * a ) ∀ a, b ᗴ R.
      So, the operation '*' is commutative over the set R. but the binary operation '/' (ordinary division) over R is not commutative for ( a /  b ) ≠ ( b / a )

3. Associative Law : Let 'A' be a non-empty set and 'O' is a binary operation defined over 'A' such that  a O ( b O c ) = ( a O b ) O c ∀ a, b, c ᗴ A.
          Then, binary operation 'O' is said to be associative over set 'A'.

4. Distributive Law : Let 'A' be a non-empty set and 'O', '*' are two binary operations defined over 'A' such that  a O  (b * c ) = ( a O b ) * ( a * c ) ∀ a, b, c ᗴ A.
         Then, the binary operation 'O' is said to be distributive over binary operation '*' in the set 'A'.

5. Existence of Identity Element : Let 'A' be a non-empty set and a binary operation 'O' is defined over 'A' for which a particular element e ᗴ A exists such that
                                                    a O e = e O a = a ∀ a ᗴ A.
        Then, we say that the identity element exists in 'A' for the binary operation 'O'.

Example :
                 For the binary operation '*' (ordinary multiplication) over the set R of real numbers, the
identity element exists and identity element is 1 for a * 1 = 1 * a = a ∀ a ᗴ R.

                For the binary operation '+' (ordinary addition) over the set of integers (Z), the identity element exists and the identity element is 0 because a + 0 = 0 + a = a ∀ a ᗴ A.

6. Existence of Inverse Element : Let 'A' be a non -empty set and a binary operation 'O' is defined over 'A' such that for each a ᗴ A there exists a corresponding element a' ᗴ A such that
           a O a' = a' O a = e, where e is the identity element of a over the operation 'O'. Then we say that for the binary operation 'O' over the set 'A', the inverse elements exists and a, a' are called inverse element of each other.

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