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

Which of the following figures has/have both line symmetry and rotational symmetry?

(i) Square                               (iii) Pentagon

(ii) Rectangle                          (iv) Isosceles triangle

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