Question

For two non-empty sets A and B, if P (A) = P (B). Show that A = B.
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Answer 1

Basic Concept :-

Power Set :-

The power set  of a Set A is defined as the set of all subsets of the Set A including the Set itself and the null or empty set. It is denoted by P(A).

Mathematically,

$\rm :\longmapsto\:If \: X \sub \: A \: then \: X \in \: P(A)$

Let's solve the problem now!!

$\rm :\longmapsto\:Let \: X \sub \: A$

$\rm :\implies\:X \in \: P(A)$

$\rm : \longmapsto\:X \in \: P(B)$

$\rm :\implies\:X \sub \: B$

$\bf\implies \:A \sub \: B \: - - (1)$

$\rm :\longmapsto\:Let \: Y \sub \: B$

$\rm :\implies\:Y \in \: P(B)$

$\rm :\longmapsto\:Y \in \: P(A)$

$\rm :\implies\:Y \sub \: A$

$\bf\implies \:B \sub \: A \: - - (2) \div$

From equation (1) and equation (2), we concluded,

$\rm :\longmapsto\:A = B$

Additional Information :-

1. Two sets A and B are equal if and only if A⊂B and B⊂A.

2. A set A is a subset of a set B, if all elements of A are also elements of B and B is then a superset of A.