Question

If A and B are two non empty sets then A∪B=B∩A iff *

A⊂B

B⊂A

A=B

None of these

​

Answer 1

Given : A and B are two non empty sets

To find : When A∪B=B∩A

A⊂B

B⊂A

A=B

None of these

Solution:

A and B are two non empty sets

Assume that A  ⊂B

=> A∪B  = B

B∩A  = A

A ≠ B

Hence A∪B ≠ B∩A

Assume that B ⊂A

=> A∪B  = A

B∩A  = B

A ≠ B

Hence A∪B ≠ B∩A

Assume that A = B

A∪B  = A  or  B

B∩A  = A  or B

A = B

Hence A∪B = B∩A

A and B are two non empty sets then A∪B=B∩A iff   A=B

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Answer 2

SOLUTION

TO CHOOSE THE CORRECT OPTION

If A and B are two non empty sets then A ∪ B = A ∩ B iff

(a) A ⊂ B

(b) B ⊂ A

(c) A = B

(d) None of these

EVALUATION

We will show that If A and B are two non empty sets then A ∪ B = A ∩ B iff A = B

First suppose that A and B are two non empty sets such that A ∪ B = A ∩ B

$\sf{Let \: \: x \: \in \: A}$

$\implies \sf{ x \: \in \: A \: \cup \: B}$

$\implies \sf{ x \in A \cap \: B} \: \: ( \because A \cup \: B = A \cap \: B)$

$\implies \sf{x \in \: B}$

$\therefore \: \: \sf{ A \: \subset \: B} \: \: .....(1)$

Again suppose that

$\sf{ y \: \in \: \: B}$

$\implies \sf{ y \: \in \: A \: \cup \: B}$

$\implies \sf{ y \in A \cap \: B} \: \: ( \because A \cup \: B = A \cap \: B)$

$\implies \sf{ y \in \: A}$

$\therefore \: \sf{B \: \subset \: A} \: \: \: ....(2)$

Hence from (1) & (2)

$\sf{A = B}$

Next suppose that A = B

Then

A ∪ B = B

A ∩ B = B

∴ A ∪ B = A ∩ B

Hence A ∪ B = A ∩ B iff A = B

FINAL ANSWER

If A and B are two non empty sets then A ∪ B = A ∩ B iff

(c) A = B

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