Question 4: Verify the commutative properties of union and intersection for the following pairs of sets:
\((ii)\) \(\;\) \(N,Z \)
Solution:
\(N=\{1,2,3,\ldots\}\)
\(Z=\{0,\pm 1, \pm 2, \pm 3, \ldots\}\)
Commutative Property Of Union: \(N\cup Z\)\(\ =\ \)\(N \cup Z\)
\[
\begin{align*}
\text{L.H.S.}&=N\cup Z\\
&=\{1,2,3,\ldots\}\cup\{0,\pm 1, \pm 2, \pm 3, \ldots\}\\
&=\{0,\pm 1, \pm 2, \pm 3, \ldots\}
\end{align*}
\]
\[
\begin{align*}
\text{R.H.S.}&=Z\cup N\\
&=\{0,\pm 1, \pm 2, \pm 3, \ldots\}\cup\{1,2,3,\ldots\}\\
&=\{0,\pm 1, \pm 2, \pm 3, \ldots\}
\end{align*}
\]
Therefore, \(N\cup Z\)\(\ =\ \)\(Z \cup N\)
Commutative Property Of Intersection: \(N\cap Z\)\(\ =\ \)\(N \cap Z\)
\[
\begin{align*}
\text{L.H.S.}&=N\cap Z\\
&=\{1,2,3,\ldots\}\cap\{0,\pm 1, \pm 2, \pm 3, \ldots\}\\
&=\{1,2,3,\ldots\}
\end{align*}
\]
\[
\begin{align*}
\text{R.H.S.}&=Z\cap N\\
&=\{0,\pm 1, \pm 2, \pm 3, \ldots\}\cap\{1,2,3,\ldots\}\\
&=\{1,2,3,\ldots\}
\end{align*}
\]
Therefore, \(N\cap Z\)\(\ =\ \)\(Z \cap N\)