Answer
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Work Step by Step
(a)
The set of all the counting numbers, namely \[1,2,3,\] and so on is called natural number. It is denoted by the letter\[N\].
\[N=\left\{ 1,2,3,4,5,... \right\}\]
Consider each element of the provided set:
Only \[\sqrt{81}\left( =9 \right)\] is a natural number.
Rest \[-17,-\frac{9}{13},0,0.75,\sqrt{2}\text{ and }\pi \] are not natural numbers.
Hence, set of natural number from the provided set is,\[\left\{ \sqrt{81} \right\}\].
(b)
The set of all the counting numbers along with \[0\], namely \[0,1,2,3,\] and so on is called whole number. It is denoted by the letter\[W\].
\[W=\left\{ 0,1,2,3,4,5,... \right\}\]
Consider each element of the provided set:
Here, \[0\text{ and }\sqrt{81}\left( =9 \right)\]are whole numbers.
Rest \[-17,-\frac{9}{13},0.75,\sqrt{2}\text{ and }\pi \] are not whole numbers.
Hence, set of whole number from the provided set is,\[\left\{ 0,\sqrt{81} \right\}\].
(c)
The set of all the counting numbers, namely \[1,2,3,\] and so on along with \[0\] and also negative of counting numbers, is called integers. It is denoted by the letter \[Z\].
\[Z=\left\{ ...,-3,-2,-1,0,1,2,3,... \right\}\]
Consider each element of the provided set:
Here,\[-17,0\text{ and }\sqrt{81}\left( =9 \right)\]are integers.
Rest \[-\frac{9}{13},0.75,\sqrt{2}\text{ and }\pi \] are not integers.
Hence, set of integers from the provided set is \[\left\{ -17,0,\sqrt{81} \right\}\].
(d)
All those numbers that can be represented in the form of \[\frac{p}{q}\], where both \[p\text{ and }q\]are integers and \[q\ne 0\] are called rational number. It is denoted by the letter \[Q\].
\[Q=\left\{ \frac{p}{q}\left| p,q\in Z\And q\ne 0 \right. \right\}\]
Consider each element of the provided set:
Here,\[-17,-\frac{9}{13},0,0.75\text{ and }\sqrt{81}\left( =9 \right)\]are rational numbers.
Rest \[\sqrt{2}\text{ and }\pi \] are not rational numbers.
Hence, set of a rational number from the provided set is \[\left\{ -17,-\frac{9}{13},0,0.75,\sqrt{81} \right\}\].
(e)
Those numbers that cannot be represented in the form of \[\frac{p}{q}\], where both \[p\text{ and }q\]are integers and \[q\ne 0\] are called irrational number. It is denoted by the letter \[I\].
\[\begin{align}
& I=R-Q \\
& =R-\left\{ \frac{p}{q}\left| p,q\in Z\And q\ne 0 \right. \right\}
\end{align}\]
Where,\[R\] is the set of Real numbers and \[Q\] is the set of Rational numbers.
Consider each element of the provided set:
Here,\[\sqrt{2}\text{ and }\pi \]are an irrational number.
Rest \[-17,-\frac{9}{13},0,0.75\text{ and }\sqrt{81}\] are not irrational numbers.
Hence, set of irrational numbers from the provided set is \[\left\{ \sqrt{2},\pi \right\}\].
(f)
The set formed by a collection of all rational and irrational numbers are called real number. It is denoted by the letter\[R\].
\[R=Q\cup I\]
Where,\[Q\] is the set of all rational numbers and \[I\] is the set of all irrational numbers.
Consider each element of the provided set:
All the elements of the provided set are real numbers
Hence, set of real numbers from the provided set is,\[\left\{ -17,-\frac{9}{13},0,0.75,\sqrt{2},\pi ,\sqrt{81} \right\}\].
