RD Sharma Solutions Class 12 Binary Operations

RD Sharma Solutions Class 12 Chapter 3

Definition: A binary operation ∗ on a nonempty set A is a function from A × A to A.

Addition, subtraction, multiplication are binary operations on Z.

Addition is a binary operation on Q because

Division is NOT a binary operation on Z because


If ∗ is a binary operation on A, an element e ∈ A is an identity element of A w.r.t

∗ if

∀a ∈ A, a ∗ e = e ∗ a = a.


Let * be a binary operation on A with identity e, and let a ∈ A. We say that a is invertible w.r.t. ∗ if there exists b ∈ A such that a ∗ b = b ∗ a = e. If f exists, we say that b is an inverse of a w.r.t. ∗ and write b = a −1 . Note, inverses may or may not exist.

Associative and Commutative Laws:

A binary operation ∗ on A is associative if

∀a, b, c ∈ A, (a ∗ b) ∗ c = a ∗ (b ∗ c).

A binary operation ∗ on A is commutative if

∀a, b ∈ A, a ∗ b = b ∗ a.

Learn in depth about the subject by working on the RD Sharma solutions class 12th Chapter 3: Binary Operations below.

Practise This Question

Suppose A1, A2,.....................A30 are thirty sets each having 5 elements and B1, B2,................Bn are n sets each with 3 elements, let 30i=1Ai=nj=1Bj=S and each element of S belongs to exactly 10 of Ai'S and exactly 9 of the Bj'S. Then n is equal to