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If $A=\{3,5,7,9,11\}, B=\{7,9,11,13\}$, $C=\{11,13,15\}$ and $D=\{15,17\}$; find
(i) $A \cap B$
(ii) $B \cap C$
(iii) $A \cap C \cap D$
(iv) $A \cap C$
(v) $B \cap D$
(vi) $A \cap(B \cup C)$
(vii) $A \cap D$
(viii) $A \cap(B \cup D)$
(ix) $(A \cap B) \cap(B \cup C)$
(x) $(A \cup D) \cap(B \cup C)$.
(i) $A \cap B$
(ii) $B \cap C$
(iii) $A \cap C \cap D$
(iv) $A \cap C$
(v) $B \cap D$
(vi) $A \cap(B \cup C)$
(vii) $A \cap D$
(viii) $A \cap(B \cup D)$
(ix) $(A \cap B) \cap(B \cup C)$
(x) $(A \cup D) \cap(B \cup C)$.
Solution:
1811 Upvotes
Verified Answer
(i) $A \cap B=\{3,5,7,9,11\} \cap\{7,9,11,13\}$
$=\{7,9,11\}$
(ii) $B \cap C=\{7,9,11,13\} \cap\{11,13,15\}=\{11,13\}$
(iii) $A \cap C \cap D$
$=\{3,5,7,9,11\} \cap\{11,13,15\} \cap\{15,17\}$
$=\{11\} \cap\{15,17\}=\phi$
(iv) $A \cap C=\{3,5,7,9,11\} \cap\{11,13,15\}=\{11\}$
(v) $B \cap D=\{7,9,11,13\} \cap\{15,17\}=\phi$
(vi) $A \cap(B \cup C)$
$\begin{aligned}
&=\{3,5,7,9,11\} \cap[\{7,9,11,13\} \\
&\cup\{11,13,15\}] \\
&=\{3,5,7,9,11\} \cap\{7,9,11,13,15\} \\
&=\{7,9,11\}
\end{aligned}$
(vii) $A \cap D=\{3,5,7,9,11\} \cap\{15,17\}=\phi$
(viii) $A \cap(B \cup D)$
$\begin{aligned}
&=\{3,5,7,9,11\} \cap[\{7,9,11,13\} \cup\{15,17\}] \\
&=\{3,5,7,9,11\} \cap\{7,9,11,13,15,17\} \\
&=\{7,9,11\}
\end{aligned}$
(ix) $(A \cap B) \cap(B \cup C)$
$\begin{aligned}
=& {[\{3,5,7,9,11\} \cap\{7,9,11,13\}] } \\
& \cap[\{7,9,11,13\} \cup\{11,13,15\}] \\
=&\{7,9,11,\} \cap\{7,9,11,13,15\} \\
=&\{7,9,11\} .
\end{aligned}$
(x)
$\begin{aligned} &(A \cup D) \cap(B \cup C) \\=& {[\{3,5,7,9,11\} \cup\{15,17\}] } \\ & \cap[\{7,9,11,13\} \cup\{11,13,15\}] \\=&\{3,5,7,9,11,15,17\} \cap\{7,9,11,13,15\} \\=&\{7,9,11,15\} . \end{aligned}$
$=\{7,9,11\}$
(ii) $B \cap C=\{7,9,11,13\} \cap\{11,13,15\}=\{11,13\}$
(iii) $A \cap C \cap D$
$=\{3,5,7,9,11\} \cap\{11,13,15\} \cap\{15,17\}$
$=\{11\} \cap\{15,17\}=\phi$
(iv) $A \cap C=\{3,5,7,9,11\} \cap\{11,13,15\}=\{11\}$
(v) $B \cap D=\{7,9,11,13\} \cap\{15,17\}=\phi$
(vi) $A \cap(B \cup C)$
$\begin{aligned}
&=\{3,5,7,9,11\} \cap[\{7,9,11,13\} \\
&\cup\{11,13,15\}] \\
&=\{3,5,7,9,11\} \cap\{7,9,11,13,15\} \\
&=\{7,9,11\}
\end{aligned}$
(vii) $A \cap D=\{3,5,7,9,11\} \cap\{15,17\}=\phi$
(viii) $A \cap(B \cup D)$
$\begin{aligned}
&=\{3,5,7,9,11\} \cap[\{7,9,11,13\} \cup\{15,17\}] \\
&=\{3,5,7,9,11\} \cap\{7,9,11,13,15,17\} \\
&=\{7,9,11\}
\end{aligned}$
(ix) $(A \cap B) \cap(B \cup C)$
$\begin{aligned}
=& {[\{3,5,7,9,11\} \cap\{7,9,11,13\}] } \\
& \cap[\{7,9,11,13\} \cup\{11,13,15\}] \\
=&\{7,9,11,\} \cap\{7,9,11,13,15\} \\
=&\{7,9,11\} .
\end{aligned}$
(x)
$\begin{aligned} &(A \cup D) \cap(B \cup C) \\=& {[\{3,5,7,9,11\} \cup\{15,17\}] } \\ & \cap[\{7,9,11,13\} \cup\{11,13,15\}] \\=&\{3,5,7,9,11,15,17\} \cap\{7,9,11,13,15\} \\=&\{7,9,11,15\} . \end{aligned}$
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