Question 4: Verify the commutative properties of union and intersection for the following pairs of sets:
\((ii)\) \(\;\) \(A=\{x|x\in R\land x\geq 0\},\)\(\;\) \(B=R \)
Solution:
\(A=\{x|x\in R\land x\geq 0\}\)
\(B=R \)
Commutative Property Of Union: \(A\cup B\)\(\ =\ \)\(A \cup B\)
\[
\begin{align*}
\text{L.H.S.}&=A\cup B\\
&=\{x|x\in R\land x\geq 0\}\cup R\\
&=R
\end{align*}
\]
\[
\begin{align*}
\text{R.H.S.}&=B\cup A\\
&=R\cup\{x|x\in R\land x\geq 0\}\\
&=R
\end{align*}
\]
Therefore, \(A\cup B\)\(\ =\ \)\(B \cup A\)
Commutative Property Of Intersection: \(A\cap B\)\(\ =\ \)\(B \cap A\)
\[
\begin{align*}
\text{L.H.S.}&=A\cap B\\
&=\{x|x\in R\land x\geq 0\}\cap R\\
&=\{x|x\in R\land x\geq 0\}
\end{align*}
\]
\[
\begin{align*}
\text{R.H.S.}&=B\cap A\\
&=R\cap\{x|x\in R\land x\geq 0\}\\
&=\{x|x\in R\land x\geq 0\}
\end{align*}
\]
Therefore, \(A\cap B\)\(\ =\ \)\(B \cap A\)