Question

For any two sets A and B prove that P(A)=P(B) implies A=B.

Answer 1

To prove that A=B we have to prove that A is the subset of B and B is the subset of A. Subset means that all the element of set A are also in set B if A is the subset of B and if B is subset of A then all elements in set B are also present in set A.

Complete step-by-step answer:
Given for any two sets, P(A)=P(B) then we have to prove that A=B.
First let A be the element of the power set P(A) because every set is a subset.
Like if P(A)=\left\\{ {\emptyset ,\left\\{ a \right\\},\left\\{ {a,b} \right\\},\left\\{ {a,b,c} \right\\}} \right\\} then A=\left\\{ a \right\\}
We can then say that A belongs to P(A)
$\Rightarrow {\text{A}} \in {\text{P(A)}}$
Now since it is given that P(A)=P(B) then we can write,
$\Rightarrow {\text{A}} \in {\text{P(B)}}$
This means that A is a subset in the power set P(B).
Then set A will also be a subset of B.
$\Rightarrow {\text{A}} \subset {\text{B}}$ --- (i)
Now let B be the element of the power set P(B) because every set is a subset.
Then we can say that B belongs to P(B)
$\Rightarrow {\text{B}} \in {\text{P(B)}}$
But given that P(A)=P(B) then we can write,
$\Rightarrow {\text{B}} \in {\text{P(A)}}$
This means that B also belongs to P(A) and B is a subset of P(A).
Then set B is also a subset of set A.
$\Rightarrow {\text{B}} \subset {\text{A}}$ --- (ii)
From (i) and (ii) we can say that
$\Rightarrow$ A=B
Hence Proved

Note: Here the student may get confused which subset symbol to use-$\subset$or $\subseteq$. We have used the $\subset$ symbol to represent the subset because it is the symbol used for Proper subset while$\subseteq$ is the symbol to represent Improper subset.
We can understand proper subsets by this example- If set A contains at least one element that is not present in set B then set A is the proper subset of set B. In a proper subset the set is not a subset of itself.
In an improper subset , the subset A contains all the elements of set B.