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.
(iii) On R, defined $$ by $a b = a b^{2}$

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Solution

On R, defined $$ by $a b = a b^{2}$
It is seen that for each $a, b \in R$ , there is a unique element $a b^{2}$ in R.
This means that $$ carries each pair (a,b) to a unique element $a b = a b^{2}$ in R. Therefore, $*$ is 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 $a * b = a b$

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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. (iii) On R, defined ∗ by a∗b=ab2

On R, 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 binary operation.

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.
(iii) On R, defined $$ by $a b = a b^{2}$

On R, defined $$ by $a b = a b^{2}$
It is seen that for each $a, b \in R$ , there is a unique element $a b^{2}$ in R.
This means that $$ carries each pair (a,b) to a unique element $a b = a b^{2}$ in R. Therefore, $*$ is binary operation.