Question

Determine whether or not each of the definitions 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 – ab.

(iii) On R, define * by a * b = ab²

(iv) On Z+, define * by a* b = |a-b|.

(v) On Z+, define * by a * b – a.

Answer 1

(i) a > b, a * b = a – b > 0 which belongs to Z+.
But if a < b, a * b = a – b < 0 which does not belong to Z+.
⇒ given operation * is not a binary operation.

(ii) For all a and b belonging to Z+, ab also belongs to Z+.
∴ The operation *, defined by a * b = ab is a binary operation.

(iii) For all a and b belonging to R, ab² also belongs to R.
⇒ The operation *, defined by a * b = ab² is a binary operation.

(iv) For all a and b belonging to Z+, |a – b| also belongs to Z+.
∴ The operation a * b = |a-b| is a binary operation.

(v) On Z+, defined by a * b = a,
a, b ∈ Z+ ⇒ a ∈ Z+.
∴ is a binary operation.