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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 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 Asked by Kumarpalaksha Last Modified 
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Answer
Azhar M
(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 = ain Z+.
Therefore, * is a binary operation.
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