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


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Answer 1

Answer:

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Step-by-step explanation:

Well, the question says  A  and  B  are non-empty ‘subsets’. Subsets of what set? Or, you may want to prove that given two non-empty ‘sets’  A  and  B , we have  A×B=B×A  if and only if  A=B . If that is the case, you can argue as follows.

Assume that  A=B . Then we have  A×B=A×A=B×A . Thus, if  A=B , we have  A×B=B×A .

Now, suppose that  A×B=B×A . To show that  A=B , we need to show that  A⊆B  and  B⊆A . So let  x∈A . Since  B  is non-empty there exists  y∈B . Thus,  (x,y)∈A×B . Since  A×B=B×A , we have  (x,y)∈B×A . Since  (x,y)∈B×A , by the definition of Cartesian product we have  x∈B . Thus,  A⊆B . Now, let  y′∈B . Since  A  is non-empty, we can  x′∈A . Thus,  (y′,x′)∈B×A=A×B , and hence  y′∈A . Therefore  B⊆A . Since  A⊆B  and  B⊆A , we have  A=B .

Answer 2

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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