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Binary Operations — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSBinary Operations2 Marks
Question
Determine whether the following operations define a binary operation on the given set or not:
$'O\ '$ on $Z$ defined by $a O b = a^b$ for all $\text{a, b}\in\text{Z.}$
✓

Answer

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

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