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Assume that P (A) = P (B). Show that A =...

Assume that `P (A) = P (B)`. Show that `A = B`

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Let x be an element of set A
then, if `x in` then `X sub A`
then, there is a subset X of set A such that `x in X`
i.e., `X sub A` such that `x in A`
`rArr X in P(A)` such that `x in X`
`rArr X in P(B)` such that `x in X [P(A)=P(B)]`
`rArr X sub B` such that `x in X`
`rArr x in B " "[because x sub X and X sub B]`
Therefore, `x in A rArr x in b`
i.e., `A sub B` ...(1)
Again, let y is any element of set B
then there is a subset y of set B such that `y in Y`
i.e., `Y sub B` such that `y in Y`
`rArr Y in P (B)` such that `y in Y`
`rArr in P(A)` such that `y in Y " "[because P(B)=P(A)]`
`rArr Y sub A` such that `y in Y`
`rArr y inA " "[because y in Y and Y sub A]`
Therefore, `y in B rArr y in A`
i.e., `B sub A` ....(2)
From equations (1) and (2), A = B.
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