Determine wherther or not each of the defintion of $$ given below gives a binary operation. In the event that $$ is not a binary operation, give justification for this.
(ii) ON $Z^{+}$ , defined $$ by $a b = a b$

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Solution

On $Z^{+}$ , $$ is defined by $a b = a b .$
It is seen that for each a,b in $Z^{+}$ , there is a unique element ab in $Z^{+} .$
This means that as $$ carries each pair (a,b)to a unique element $a b = a b i n Z^{+} .$
Therefore, $*$ is a binary operation.

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Determine wherther or not each of the defintion of ∗ given below gives a binary operation. In the event that ∗ is not a binary operation, give justification for this. (ii) ON Z+, defined ∗ by a∗b=ab

On Z+, ∗ is defined by a∗b=ab. It is seen that for each a,b in Z+, there is a unique element ab in Z+. This means that as ∗ carries each pair (a,b)to a unique element a∗b=ab in Z+. Therefore, ∗ is a binary operation.

Determine wherther or not each of the defintion of $$ given below gives a binary operation. In the event that $$ is not a binary operation, give justification for this.
(ii) ON $Z^{+}$ , defined $$ by $a b = a b$

On $Z^{+}$ , $$ is defined by $a b = a b .$
It is seen that for each a,b in $Z^{+}$ , there is a unique element ab in $Z^{+} .$
This means that as $$ carries each pair (a,b)to a unique element $a b = a b i n Z^{+} .$
Therefore, $*$ is a binary operation.