theorems prove these results in the more general form. Theorem . : (Uniqueness of Identity) In an algebraic structure the identity element (if exists) must be unique. Proof Let ( , ) S ∗ be an algebraic structure.
Assume that the identity element of S exists in S . It is to be proved that the identity element is unique. Suppose that e and e be any two identity elements of S . First treat e as the identity and e as an arbitrary element of S .
Then by the existence of identity property, e ∗ ∗ ... ( ) Interchanging the role of e and e , e ∗ ∗ …( ) From ( ) and ( ), e . Hence the identity element is unique which completes the proof. Theorem .
(Uniqueness of Inverse) In an algebraic structure the inverse of an element (if exists) must be unique. Proof Let ( , ) S ∗ be an algebraic structure and a S . Assume that the inverse of a exists in S . It is to be proved that the inverse of a is unique.
The existence of inverse in S ensures the existence of the identity element e in S . - - Discrete Mathematics Let a S . It is to be proved that the inverse a (if exists) is unique. Suppose that a has two inverses, say, a , a .
Treating a as an inverse of a gives a a ∗ ∗ …( ) Next treating a as the inverse of a gives a a ∗ ∗ …( ) a a e a ∗= ∗ ∗ ∗ ∗ = ∗ (by ( ) and ( )). So, a . Hence the inverse of a is unique which completes the proof. Example .
Verify the (i) closure property, (ii) commutative property, (iii) associative property (iv) existence of identity and (v) existence of inverse for the arithmetic operation + on . (i) m ∈ , ∀ m n . Hence + is a binary operation