Question

Show that A ∪ B = A ∩ B implies A = B

Answer 1

if AUB=AIntersection B
then all the elements in A are said to be in B as well.
that implies A=B

Answer 2

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Let A = B

AUB = A 

A intersection B = A implies A U B  = A intersection B

A = B implies AU B = intersection B ---- (1)

Now let A U B  = A intersection B. Then we have to prove that A = B. For this let x belongs to A implies x belongs to A U B

= x belongs to A intersection B

= x belongs to A and x Belongs to B

= x belongs to B

so A subset of B --- (2)

Now we will let y belong to B which implies y belongs to A U B 

= y belongs to A intersection B

= y belongs to A and y belongs to B

= y belongs to A

Therefore, B subset of A --- (3)

from (2) and (3) we get A=B

so  A union B will be equal to A intersection B implies A = B ---(4)

From (1) and (4) we will get  

A union B = A INTERSECTION B Implies A = B