Determine whether the following operations define a binary operat… - Vidyadip

Question

Determine whether the following operations define a binary operation on the given set or not:
${ }^2{ }^{\prime}$ ' on $N$ defined by $a^* b=a^b$ for all $a, b \in N$.

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Answer

Let $\text{a, b}\in\text{N}$
Then, $\text{a}^{\text{b}}\in\text{N}$ $\big[$Therefore $\text{a}^{\text{b}}\neq0$ and $a^b$ is positive integer$\big]$
Implies that $\text{a}\ ^*\ \text{b}\in\text{N}$
Therefore, $\text{a}\ ^*\ \text{b}\in\text{N},\ \forall\ \text{a, b}\in\text{N}$
Thus, * is a binary operation on N.

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