Question

if AUB=AnB
Then prove that A=B​

Answer 1

Question :-

If  A ∪ B = A ∩ B

then prove that A = B .

Solution :-

Let,

x ∈ A

then

x ∈   A ∪ B

now, ∵ A ∪ B = A ∩ B

∴ x ∈ A ∩ B

it means

x ∈ B  

so, x ∈ A and x ∈ B ,

it means if an element belongs to set A then , it must belong to set B also,

∴    A ⊂ B  ......eqn (1)

now similarly,

Let,

y ∈ B

then

y ∈ A ∪ B

∵ A ∪ B = A ∩ B

∴   y ∈ A ∩ B

it means

y ∈ A

so, y ∈ B and y ∈ A ,

it means if any element belongs to set B then it will also belong to set A

∴ B ⊂ A ......eqn(2)

By eqn(1) and (2) we know,

A ⊂ B  and  B ⊂ A

therefore,

A = B .

proved.

: )