Show that A B = A B implies A = B - Vidyadip

Question

Show that A $\cup$ B = A $\cap$ B implies A = B

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Answer

Suppose a $\in$ A.
Then a $\in$ A $\cup$ B. Since A $\cup$ B = A $\cap$ B , a $\in$ A $\cap$ B. So a $\in$ B.
Thus,A ⊂ B. Similarly, if b ∈ B, then b ∈ A ∪ B.
A $\cup$ B = A $\cap$ B, b $\in$ A $\cap$ B. So, b $\in$ A.
Thus, B $\subset$ A. $\Rightarrow$ A = B

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