Question

For any two sets A and B,prove that
P(A)=P(B)=A=B​

Answer 1

Hint: 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

Explanation:

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)={∅,{a},{a,b},{a,b,c}}{∅,{a},{a,b},{a,b,c}} then A={a}{a}

We can then say that A belongs to P(A)

⇒A∈P(A)⇒A∈P(A)

Now since it is given that P(A)=P(B) then we can write,

⇒A∈P(B)⇒A∈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.

⇒A⊂B⇒A⊂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)

⇒B∈P(B)⇒B∈P(B)

But given that P(A)=P(B) then we can write,

⇒B∈P(A)⇒B∈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.

⇒B⊂A⇒B⊂A --- (ii)

From (i) and (ii) we can say that

⇒⇒ A=B

Hence Proved

Note: Here the student may get confused which subset symbol to use-⊂⊂or ⊆⊆. We have used the ⊂⊂ symbol to represent the subset because it is the symbol used for Proper subset while⊆⊆ 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.