Binary Operation & Algebric Structure

Let G be a non-empty set then an operation ‘o’ on the non-empty set G is called binary operation.

If a∊G, b∊G ⇒ aob∊G, ∀ a, b∊G

This property is called closure property and if this is satisfied then G is said to be closed under the binary composition ‘o’.

Example:- Addition operation (+) as well as multiplication operation on the set N (natural number)

a∊N, b∊N ⇒ a+b∊N, ∀ a,b∊N

& a∊N, b∊N ⇒ a.b∊N, ∀ a,b∊N

A non-empty set G along with one or more binary operation or composition on ‘G’ is called an Algebraic System.

Thus each of the system (N,+) , (Q,+,.) , (R,+,.) , {ρ(A),⋃,⋂} are the algebraic system.

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