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) OnZ +, 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.