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.

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 = 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
(i) On Z⁺, * is defined by a * b = a − b.
It is not a binary operation as the image of (1, 2) under * is 1 * 2 = 1 − 2 = −1 ∉ Z⁺.
(ii) On Z⁺, * is defined by a * b = ab.
It is seen that for each a, b ∈ Z⁺, there is a unique element ab in Z⁺.
This means that * carries each pair (a, b) to a unique element a * b = ab in Z⁺. Therefore, * is a binary operation.
(iii) On R, * is defined by a * b = ab².
It is seen that for each a, b ∈ R, there is a unique element ab² in R.
This means that * carries each pair (a, b) to a unique element a * b = ab² in R. Therefore, * is a binary operation.
(iv) On Z⁺, * is defined by a * b = |a − b|.
It is seen that for each a, b ∈ Z⁺, there is a unique element |a − b| in Z⁺. This means that * carries each pair (a, b) to a unique element a * b = |a − b| in Z⁺.
Therefore, * is a binary operation.
(v) On Z⁺, * is defined by a * b = a.
* carries each pair (a, b) to a unique element a * b = a in Z⁺. Therefore, * is a binary operation.