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If $A=\{x: x$ is a natural number $\}$,
$B=\{x: x$ is an even natural number $\}$
$C=\{x: x$ is an odd natural number $\}$ and
$D=\{x: x$ is a prime number $\}$, find
(i) $A \cap B$
(ii) $A \cap C$
(iii) $A \cap D$
(iv) $B \cap C$
(v) $B \cap D$
(vi) $C \cap D$
$B=\{x: x$ is an even natural number $\}$
$C=\{x: x$ is an odd natural number $\}$ and
$D=\{x: x$ is a prime number $\}$, find
(i) $A \cap B$
(ii) $A \cap C$
(iii) $A \cap D$
(iv) $B \cap C$
(v) $B \cap D$
(vi) $C \cap D$
Solution:
2926 Upvotes
Verified Answer
Given, $A=\{1,2,3,4, \ldots \}$.
$B=\{2,4,6,8, \ldots .\}$.
$C=\{1,3,5,7, \ldots\}$
and $D=\{2,3,5,7,11,13, \ldots\}$.
(i) $A \cap B=\{1,2,3,4, \ldots\} \cap\{2,4,6,8, \ldots\}$
$=\{2,4,6,8, \ldots\}=B$.
(ii) $A \cap C=\{1,2,3,4, \ldots\} \cap\{1,3,5,7\}$
$=\{1,3,5,7, \ldots\}=C$
(iii) $A \cap D=\{1,2,3,4, \ldots\} \cap\{2,3,5,7,11,13, \ldots\}$
$=\{2,3,5,7,11,13, \ldots\}=D$
(iv) $B \cap C=\{2,4,6,8, \ldots\} \cap\{1,3,5,7, \ldots\}=\phi$
(v) $B \cap D=\{2,4,6,8, \ldots\} \cap\{2,3,5,7,11,13, \ldots\}=.\{2\}$
(vi) $C \cap D=\{1,3,5,7, \ldots\} \cap\{2,3,5,7,11,13, \ldots\}$.
$=\{3,5,7,11,13, \ldots\}$
$=\{x: x$ is an odd prime number $\}$.
$B=\{2,4,6,8, \ldots .\}$.
$C=\{1,3,5,7, \ldots\}$
and $D=\{2,3,5,7,11,13, \ldots\}$.
(i) $A \cap B=\{1,2,3,4, \ldots\} \cap\{2,4,6,8, \ldots\}$
$=\{2,4,6,8, \ldots\}=B$.
(ii) $A \cap C=\{1,2,3,4, \ldots\} \cap\{1,3,5,7\}$
$=\{1,3,5,7, \ldots\}=C$
(iii) $A \cap D=\{1,2,3,4, \ldots\} \cap\{2,3,5,7,11,13, \ldots\}$
$=\{2,3,5,7,11,13, \ldots\}=D$
(iv) $B \cap C=\{2,4,6,8, \ldots\} \cap\{1,3,5,7, \ldots\}=\phi$
(v) $B \cap D=\{2,4,6,8, \ldots\} \cap\{2,3,5,7,11,13, \ldots\}=.\{2\}$
(vi) $C \cap D=\{1,3,5,7, \ldots\} \cap\{2,3,5,7,11,13, \ldots\}$.
$=\{3,5,7,11,13, \ldots\}$
$=\{x: x$ is an odd prime number $\}$.
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