Question
Determine whether or not each of the definition of * given below gives a binary operation. In the event that * is not a binary operation, give justification for this
(i) On Z+, define * by a * b = a – b (ii) On Z+, define *by a * b = a b
(iii) On R , define * by a * b = a b2 (iv) On Z+, define * by a * b = | a – b |
(v) On Z+ , define * by a * b = a
Solution
(i) If a, b ∈ Z+, then a – b may or may not belong to H ; for example 3–5 = –2 ∉ Z. is not a binary operation on Z+.
(ii) If a, b ∈ Z+, then a b also belong to Z+ . is a binary operation on Z+.
(iii) If a, b ∈ R, then a b2 ∈ R.
(iv) Since | a – b | ≥ 0 , therefore, for all a, b ∈ Z+, a * b ∈ Z+. is a binary operation on Z+.
(v) For all a, b ∈ Z+ a * b = a ∈ Z+, is a binary operation on Z+.
