Question:
Let $A=\{x: x \in N\}, B=\{x: x-2 n, n \in N\}, C=\{x: x=2 n-1, n \in N\}$ and $D=\{x: x$ is a prime natural 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:
$A=\{x: x \in N\}=\{1,2,3, \ldots\}$
$B=\{x: x-2 n, n \in N\}=\{2,4,6,8, \ldots\}$
$C=\{x: x=2 n-1, n \in N\}=\{1,3,5,7, \ldots\}$
D = {x:x is a prime natural number.} = {2, 3, 5, 7,...}
(i) $A \cap B=B$
(ii) $A \cap C=C$
(iii) $A \cap D=D$
(iv) $B \cap C=\phi$
(v) $B \cap D=\{2\}$
(vi) $C \cap D=D-\{2\}$