Determine whether or not the definition of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation,give justification for this. On $R$,define $*$ by $a * b = ab^2$.
(A) On $R$,the operation $*$ is defined by $a * b = ab^2$.
For any two real numbers $a, b \in R$,the product $ab^2$ is also a real number because the set of real numbers $R$ is closed under multiplication.
Since for every pair $(a, b) \in R \times R$,there exists a unique element $ab^2 \in R$,the operation $*$ satisfies the definition of a binary operation.
Therefore,$*$ is a binary operation on $R$.
For any two real numbers $a, b \in R$,the product $ab^2$ is also a real number because the set of real numbers $R$ is closed under multiplication.
Since for every pair $(a, b) \in R \times R$,there exists a unique element $ab^2 \in R$,the operation $*$ satisfies the definition of a binary operation.
Therefore,$*$ is a binary operation on $R$.
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