Question

if A and B are two non empty sets then AUB and AnB iff

a. A<B
b. B<A
c. A=B
d. none of these ​

Answer 1

Given : A and B are two non empty sets then AUB =  A∩B iff  

To Find  : Correct option

a. A⊂B

b. B⊂A

c. A=B

d. none of these

Solution:

AUB =  A∩B

iff  A  = B

as A = B  then A∩B  = A = B

AUB = A = B  

=>  AUB =  A∩B = A = B

if A ⊂ B

Then  AUB = B

A ∩ B = A

Hence AUB ≠ A ∩ B

AUB =  A ∩ B if A = B

if B ⊂ A

Then  AUB = A

A ∩ B = B

Hence AUB ≠ A ∩ B

AUB =  A ∩ B if A = B

Correct option is  A = B

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Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

TO CHOOSE THE CORRECT OPTION

A and B are two non empty sets then

$\sf{A \cup B = A \cap B \: } \: \: \: iff$

$\sf{ a. \: \: \:A \subset \: B \: \: }$

$\sf{b. \: \: \: \: B \subset \: A \: }$

$\sf{ c. \: \: \:A = \: B \: \: }$

$\sf{ d. \: \: none \: of \: these \: \: }$

ANSWER

A and B are two non empty sets then

$\sf{A \cup B = A \cap B \: } \: \: \: iff$

$\sf{ c. \: \: \:A = \: B \: \: }$

EXPLANATION

$\sf{First \: suppose \: that \: \: A = B}$

Then

$\sf{A \cup B = A \: } \:$

and

$\sf{ A \cap B \: = A} \: \: \:$

So

$\sf{A \cup B = A \cap B \: } \: \: \:$

Hence

$\sf{ A = B\: \: \: implies \: \: \: \: A \cup B = A \cap B \: } \: \: \:$

$\sf{Next \: suppose \: that \: \: \: \: A \cup B = A \cap B \: }$

Then

$\sf{A } \: \: \:$

$= \sf{A \cap(A \cup B) \: } \: \: \: ( By \: Absorptive \: Property \: ) \: \:$

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

$= \sf{A \cap B \: } \: \:$

$= \sf{B \: \cap \: A }$

$= \sf{B \cap \: (B \cap A) \: } \: \: \: \:$

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

$= \sf{B } \: \: \: \: ( By \: Absorptive \: Property \: )$

So

$\sf{A \cup B = A \cap B \: } \: \: \:implies \: \: \: \: A = B$

Therefore

$\sf{A \cup B = A \cap B \: } \: \: \:iff \: \: \: \: A = B$

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