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
∀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.