Question

Is it true that for any sets A and B, P (A) ∪ P (B) = P (A ∪ B)? Justify your answer.

Answer 1

Let’s assume,

A={ 1,2 }

B={ 2,3 }

The power set of A is,

P( A )={ { 1 },{ 2 },{ 1,2 },ϕ }

The power set of B is,

P( B ) = { { 2 }, { 3 }, { 2, 3 }, ϕ }

P( A )∪P( B )={ { 1 },{ 2 },{ 1,2 },{ 3 },{ 2,3 },ϕ }(1)

Now,

A∪B={ 1,2,3 }

The power set of A∪B  is,

P( A∪B )={ { 1 },{ 2 },{ 3 },{ 1,2 },{ 1,3 },{ 2,3 },{ 1,2,3 },ϕ }(2)

Compare equations (1) and (2).

P( A )∪P( B )≠P( A∪B )

Hence, the statement is false.