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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