Find Intersections of Sets A, B, C and D
Let \[ A=\{x:x \in N\} \] \[ B=\{x:x=2n,\ n \in N\} \] \[ C=\{x:x=2n-1,\ n \in N\} \] \[ D=\{x:x \text{ is a prime natural number}\} \] 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
Since \[ B,C,D \subseteq A \]
(i) \[ A \cap B=B \]
(ii) \[ A \cap C=C \]
(iii) \[ A \cap D=D \]
(iv) \[ B \cap C=\Phi \] because no number is both even and odd.
(v) \[ B \cap D=\{2\} \] because \[ 2 \] is the only even prime number.
(vi) \[ C \cap D=\{3,5,7,11,\ldots\} \] that is, all odd prime numbers.