Question 4: Verify the commutative properties of union and intersection for the following pairs of sets:
\((i)\) \(\quad\) \(A=\{1,2,3,4,5\},\)\(\;\)\(B=\{4,5,6,10\}\)
Solution:
Commutative Property Of Union: \(A\cup B\)\(\ =\ \)\(B \cup A\)
\[
\begin{align*}
\text{L.H.S.}&=A\cup B\\
&=\{1,2,3,4,5\}\cup\{4,5,6,10\}\\
&=\{1,2,3,4,5,6,10\}
\end{align*}
\]
\[
\begin{align*}
\text{R.H.S.}&=B\cup A\\
&=\{4,5,6,10\}\cup\{1,2,3,4,5\}\\
&=\{1,2,3,4,5,6,10\}
\end{align*}
\]
Therefore, \(A\cup B\)\(\ =\ \)\(B \cup A\)
Commutative Property Of Intersection: \(A\cap B\)\(\ =\ \)\(B \cap A\)
\[
\begin{align*}
\text{L.H.S.}&=A\cap B\\
&=\{1,2,3,4,5\}\cap\{4,5,6,10\}\\
&=\{4,5\}
\end{align*}
\]
\[
\begin{align*}
\text{R.H.S.}&=B\cap A\\
&=\{4,5,6,10\}\cap\{1,2,3,4,5\}\\
&=\{4,5\}
\end{align*}
\]
Therefore, \(A\cap B\)\(\ =\ \)\(B \cap A\)