Question

If A is any set and P (A) is its power set, then which of the following is/are correct?
 I. \[P\left( A \right) \cap P\left( B \right) = P(A \cap B)\]
II. \[P(A) \cup P(B) = P(A \cup B)\]
Select the correct answer using the code given below.
A.Only I
B.Only II
C.Both I and II
D.Neither I nor II

Answer 1

Hint: Here, we will consider any two sets, make their power sets and according to the question, find the power sets of their intersection and union as well to get the final answer.
‘P’ denotes the respective power sets which means the combination of all possible subsets of any set.

Complete step-by-step answer:
Let the sets be:
A = {p,q,r} B = {r,s}
Intersection of sets (common members):
 $ A \cap B = \{ r\} $
Union of sets (combined members, without repeating the common):
 $ A \cup B = \{ p,q,r,s\} $
The sets will be equal only when the number of members and the members will be the same.
Power sets of A and B:
 $ P(A) = \{ \phi ,p,q,r,\{ p,q\} ,\{ q,r\} ,\{ r,p\} ,\{ p,q,r\} \} $
 $ P(B) = \{ \phi ,r,s,\{ r,s\} \} $
—Checking for statement I:
 $ P(A) \cap P(B) = \{ \phi ,r\} $
 $ P(A \cap B) = \{ \phi ,r\} $
Both of them are equal; this shows that statement I.
\[P\left( A \right) \cap P\left( B \right) = P(A \cap B)\]is correct.
Checking for statement II.
 $ P(A) \cup P(B) = \{ \phi ,p,q,r,\{ p,q\} ,\{ q,r\} ,\{ r,p\} ,\{ r,s\} ,\{ p,q,r\} \} $
But for $ P(A \cup B) $ , the power set will contain 16 members as the set has 4 members and $ {2^4} = 16 $ .
As the number sets will not be equal for both power sets, this means they won’t be equal either.
This shows that statement II.
II. \[P\left( A \right) \cup P\left( B \right) = P(A \cup B)\]is incorrect.
So, the correct answer is “Option A”.

Note: If the original set has n members then the power set will have $ {2^n} $ members.
 $ '\phi ' $ which denotes an empty or a null set is always a member of the power set and subset of any set.