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Question
Determine whether or not the definition of * given below gives a binary operation. In the event that * is not a binary operation give justification of this.
On R, define * by a * b = a + 4b2
Here, Z+ denotes the set of all non-negative integers.
Solution
\[ a, b \in R\]
\[ \Rightarrow a, 4 b^2 \in R\]
\[ \Rightarrow a + 4 b^2 \in R\]
\[ \Rightarrow a * b \in R\]
\[\text{Therefore},\]
\[a * b \in R, \forall a, b \in R\]
\[\text{Thus}, * \text{ is a binary operation on R }.\]
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