Question

If A = {x: x is a natural number}, B = {x: x is an even number}, C = {x: x is an odd 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

Answer 1

Hint: We will first write the roster form of the sets A, B, C and D. Then, we will solve every part one by one using the roster form of sets. We need to find the intersection of two different sets and we know that intersection of two given sets is defined as the largest set which contains all the elements common to both the sets.

Complete step-by-step answer:
We are given four sets defined as:
A = {x: x is a natural number}
B = {x: x is an even number}
C = {x: x is an odd number}
D = {x: x is a prime number}
Let us write all the sets in roster form as:
A = {x: x is a natural number}
    = {1, 2, 3, 4, 5, …}
B = {x: x is an even number}
    = {2, 4, 6, 8, 10, …}
C = {x: x is an odd number}
    = {1, 3, 5, 7, 9, …}
D = {x: x is a prime number}
    = {2, 3, 5, 7, 11, …}
Now, let us consider the parts of this question:
(i)A\[ \cap \]B
By definition of intersection of sets, we get
$ \Rightarrow A \cap B = \left\{ {1,2,3,4,5,...} \right\} \cap \left\{ {2,4,6,8,10,...} \right\}$
 = {2, 4, 6, 8, 10, …}
 = B

(ii)A \[ \cap \]C
$ \Rightarrow $A \[ \cap \]C = {1, 2, 3, 4, 5, …}\[ \cap \]{1, 3, 5, 7, 9, …}
= {1, 3, 5, 7, 9, …}
= C

(iii)A\[ \cap \]D
 $ \Rightarrow $A\[ \cap \]D = {1, 2, 3, 4, 5, …}\[ \cap \]{2, 3, 5, 7, 11, …}
   = {2, 3, 5, 7, 11}
  = D

(iv)B\[ \cap \]C
$ \Rightarrow $B\[ \cap \]C = {2, 4, 6, 8, 10, …}\[ \cap \]{1, 3, 5, 7, 9, …}
Here, we can see that there is no common element as one is a set of even numbers and the other is a set of odd numbers. Therefore, B\[ \cap \]C is a null set.
$ \Rightarrow $B\[ \cap \]C = {}
= $\phi $

(v)B\[ \cap \]D
$ \Rightarrow $B\[ \cap \]D = {2, 4, 6, 8, 10, …}\[ \cap \]{2, 3, 5, 7, 11, …}
= {2}

(vi)C\[ \cap \]D
$ \Rightarrow $C\[ \cap \]D = {1, 3, 5, 7, 9, …}\[ \cap \]{2, 3, 5, 7, 11}
= {1, 3, 5, 7, 11, …}
This set can also be written in set – builder form as: C\[ \cap \]D = {x: x is an odd prime number}

Note: In such questions, you may go wrong while calculating the roster form of the sets from the set – builder form. Also, you must be very careful while finding the intersection of given two sets. You can also solve this question using the definitions of the natural numbers, odd / even numbers and prime numbers.