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 = ab
(iii) On R, define * by a * b = ab^2
(iv) On Z + , define * by a * b = |a − b|
(v) On Z + , define * by a * b = a

Answer 1

(i)  given that On Z⁺, define ∗ 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) given that On Z⁺, define *by a * b = ab
We can see that for each a, b ∈ Z+, there is a unique element ab in Z⁺.
⇒ * carries each pair (a, b) to a unique element a * b = ab in Z⁺. Therefore, * is a binary operation.

(iii) given that On R, define ∗ by a ∗ b = ab²
We can see that for each a, b ∈  R, there is a unique element ab² in R.
⇒ * carries each pair (a, b) to a unique element a * b = ab² in R.
therefore, * is a binary operation.

(iv) given On Z⁺, define ∗ by a ∗ b = | a – b|
We can see that for each a, b ∈ Z⁺, there is a unique element |a – b| in Z⁺.
⇒ * carries each pair (a, b) to a unique element
a * b = |a – b| in Z+. Therefore, * is a binary operation.

(v) given that On Z⁺ define ∗ by a ∗ b = a
We can see that for each a, b ∈ Z⁺, there is a unique element a in Z⁺.
⇒ * carries each pair (a, b) to a unique element a * b = a in Z⁺.Therefore, * is a binary operation.

Answer 2

Answer:

(i)  given that On Z⁺, define ∗ 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) given that On Z⁺, define *by a * b = ab

We can see that for each a, b ∈ Z+, there is a unique element ab in Z⁺.

⇒ * carries each pair (a, b) to a unique element a * b = ab in Z⁺. Therefore, * is a binary operation.

(iii) given that On R, define ∗ by a ∗ b = ab²

We can see that for each a, b ∈  R, there is a unique element ab² in R.

⇒ * carries each pair (a, b) to a unique element a * b = ab² in R.

therefore, * is a binary operation.

(iv) given On Z⁺, define ∗ by a ∗ b = | a – b|

We can see that for each a, b ∈ Z⁺, there is a unique element |a – b| in Z⁺.

⇒ * carries each pair (a, b) to a unique element

a * b = |a – b| in Z+. Therefore, * is a binary operation.

(v) given that On Z⁺ define ∗ by a ∗ b = a

We can see that for each a, b ∈ Z⁺, there is a unique element a in Z⁺.

⇒ * carries each pair (a, b) to a unique element a * b = a in Z⁺.Therefore, * is a binary operation.