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:
' O ' on Z defined by a $Ob = a ^{ b }$ for $all a , b \in Z$.

✓

Answer

We have,
$a$ O $b=a^b$ for all $a, b \in Z$
Let $a \in Z$ and $b \in Z$
$\Rightarrow a^{b} \notin Z \Rightarrow aOb \notin Z$
For example, if $a=2, b=-2$
$\Rightarrow a^{b}=2^{-2}=\frac{1}{4} \notin Z$
$\therefore$ The operation 'O' does not define a binary operation on Z .

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