Question

Given A, B and C are sets, consider the following text:

(1) If B ∩ C = ∅ and A ⊂ (B ∪ C) then (A ∪ B) ∩ C = A ∩ B

(2) A ∪ (B ∩ C) ⊂ (A ∪ C) ∩ B

(3) If set A has 9 elements, set B has 7 elements and power set of A - B has 32 elements then power set of B - A has 16 elements.


Which statements are correct?

A. (1) and (2) are correct but (3) is not.

B. (1) and (3) are correct but (2) is not.

C. (2) and (3) are correct but (1) is not.

D. All are correct.

E. All are incorrect.

Answer 1

Let's check one by one

#1

• B$\cap$ C=Ø

And

• A$\subset$ (B U C)
• A$\subset$ B U A$\subset$ C

Till now it's ok but

• (A UB)$\cap$ C
• A$\cap$ C U B$\cap$C

But as B$\cap$ C has no elements this expression yields A$\cap$ C

Hence

• False

#2

Solve RHS

• (A UC )$\cap$ B
• A$\cap$ B U B$\cap$ C

But according to first statement second part has no elements so it yields only

• A$\cap$ B

And

It denotes to common elements of A and B not all elements of A

Hence

• False

#3

$\\ \rm\Rrightarrow P(A-B)=32$

$\\ \rm\Rrightarrow 2^{|A-B|}=32$

$\\ \rm\Rrightarrow 2^{|A-B|}=2^5$

$\\ \rm\Rrightarrow |A-B|=5$

$\\ \rm\Rrightarrow A-B=\pm 5$

As elements can't be negative integer it's only 5.

$\\ \rm\Rrightarrow A\cap B=A-(A-B)=9-5=4\star$

And

$\\ \rm\Rrightarrow P(B-A)=16$

$\\ \rm\Rrightarrow 2^{|B-A|}=16$

$\\ \rm\Rrightarrow 2^{|B-A|}=2^4$

$\\ \rm\Rrightarrow |B-A|=4$

$\\ \rm\Rrightarrow B-A=\pm 4$

Again take it positive 4

$\\ \rm\Rrightarrow B\cap A=B-(B-A)=7-4=3$

As

$\\ \rm\Rrightarrow A\cap B\neq B\cap A$

• This statement is false

Conclusion:-

All statements are incorrect

• Option E