# Binary Operations

Binary Operations

The binary operation can be defined as an operation * which is performed on a set A. The function is given by *: A * A → A. So the operation * performed on operands a and b is denoted by a * b.

(i) A binary operation * on a set A is a function * : A × A → A. We denote * (a, b) by a * b.

(ii) A binary operation * on the set P is called commutative, if a * b = b * a for every a, b ∈P.

(iii) A binary operation * : A × A → A is said to be associative if (a * b) * c = a * (b * c), for every a, b, c ∈ A.

(iv) Given a binary operation * : A × A → A, an element e ∈ A, if it exists, is called identity for the operation *, if a * e = a = e * a, ∀ a ∈ A.

(v) Given a binary operation * : A × A → A, with the identity element e in A, an element a ∈ A, is said to be invertible with respect to the operation *, if there exists an element b in A such that a * b = e = b * a and b is called the inverse of a and is denoted by a–1.

Example  Is the binary operation * defined on Z (set of integer) by
m * n = m – n + mn ∀ m, n ∈ Z commutative?
Solution No. Since for 1, 2 ∈ Z, 1 * 2 = 1 – 2 + 1.2 = 1 while 2 * 1 = 2 – 1 + 2.1 = 3
so that 1 * 2 ≠ 2 * 1