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 = ab²
(iv) On Z⁺, define * by a * b = |a − b|
(v) On Z⁺, define * by a * b = a

Answer 1

 (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⁺.

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(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.

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(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.

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(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.

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(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.