Binary Operations

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  1. *, a binary operation on a set A is basically a function *: A × A → A. We denote * (a, b) by a * b.
  2. if a * b = b * a for every a, b ∈ X, a binary operation * on the set X is called commutative.
  3. A binary operation *: A × A → A is said to be associative if (a * b) * c = a * (b * c), for every a, b, c ∈ A.
  4. Given a binary operation *: A × A → A, an element e ∈ A, if a * e = a = e * a, ∀ a ∈ A, is called identity for the operation *.
  5. Given a binary operation *: A × A → A, with the identity element e in A,with respect to the operation * an element a ∈ A, is said to be invertible, if there exists an element b in A such that a * b = e = b * a. Here b is called the inverse of a and is denoted by a –1.

Example: Let * be a binary operation defined on Q. Point out which of the following binary operations are associative

  1. a * b = a – b for a, b ∈ Q.
  2. a * b = 4 ab for a, b ∈ Q.
  3. a * b = a – b + ab for a, b ∈ Q.

Solution

i)if we take a = 1, b = 2 and c = 3, then

(a * b) * c = (1 * 2) * 3 = (1 – 2) * 3 = – 1 – 3 = – 4 and

a * (b * c) = 1 * (2 * 3) = 1 * (2 – 3) = 1 – ( – 1) = 2.

Thus (a * b) * c ≠ a * (b * c) and hence * is not associative.

(ii) * is associative since Q is associative with respect to multiplication.

(iii) * is not associative for if we take a = 2, b = 3 and c = 4, then (a * b) * c = (2 * 3) * 4 = (2 – 3 + 6) * 4 = 5 * 4 = 5 – 4 + 20 = 21, and a * (b * c) = 2 * (3 * 4) = 2 * (3 – 4 + 12) = 2 * 11 = 2 – 11 + 22 = 13 Thus (a * b) * c ≠ a * (b * c) and hence * is not associative.