Determine whether or not the definition of * given below gives a … - Vidyadip

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 $a * b = a + 4b^2$
Here, $Z^+$ denotes the set of all non-negative integers.

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Answer

$\text{a, b}\in\text{R}$$\Rightarrow\ \text{a, 4b}^2\in\text{R}$
$\Rightarrow\ \text{a}+\text{4b}^2\in\text{R}$
$\Rightarrow\ \text{a}\ ^*\ \text{b}\in\text{R} $
Therefore,
$\text{a}\ ^*\ \text{b}\in\text{R},\ \forall\ \text{a, b}\in\text{R}$
Thus, $*$ is a binary operation on $R.$

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