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# Determine whether or not each of the definition of $\ast$ given below gives a binary operation. In the event that $\ast$ is not a binary operation, give justification for this: On $R$,defined $\ast\;by\;a\;\ast \;b=ab^2$

Note: This is part 3 of a 5 part question, split as 5 separate questions here.

Toolbox:
• A binary operation $∗$ on a set $A$ is a function $∗$ from $A \times A$ to $A$. Therefore, if $a,b \in A \Rightarrow a*b \in A\; \forall\; a,b, \in A$
Given On $R$ defined $*$ by $a*b=ab^2$
$\Rightarrow$ For each pair of element $a,b \in R$, there must exist an unique element $ab^2 \in R \rightarrow *$ defined $(a,b)$ carries unique element $a*b=ab^2 \in R$
Therefore it is a binary operation.
edited Mar 19, 2013