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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 − b

(ii) On Z+, define * by ab

(iii) On R, define * by ab2

(iv) On Z+, define * by = |− b|

(v) On Z+, define * by a

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(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...
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(i) On Z+, * is defined by * 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 ab ∈ Z+, there is a unique element ab in Z+.

This means that * carries each pair (ab) to a unique element * b ab in Z+.

Therefore, * is a binary operation.

(iii) On R, * is defined by a * b = ab2.

It is seen that for each ab ∈ R, there is a unique element ab2 in R.

This means that * carries each pair (ab) to a unique element * b abin R.

Therefore, * is a binary operation.

(iv) On Z+, * is defined by * b = |a − b|.

It is seen that for each ab ∈ Z+, there is a unique element |a − b| in Z+.

This means that * carries each pair (ab) to a unique element * b =
|a − b|in Z+.

Therefore, * is a binary operation.

(v) On Z+, * is defined by a * b = a.

* carries each pair (ab) to a unique element * b ain Z+.

Therefore, * is a binary operation.

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