Let A and B be sets. Prove that A = B if and only if A ∪ B = A ∩ B


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Question Number 44826 by Joel578 last updated on 05/Oct/18

$Let A and B be sets .$ $Prove that A = B if and only if A \cup B = A \cap B$
	Answered by Kunal12588 last updated on 05/Oct/18

	$A \cup B = A \cap B . . . . . . (1)$ $l e t x \in A$ $\Rightarrow x \in A \cup B$ $\Rightarrow x \in A \cap B f r o m (1)$ $\Rightarrow x \in A a n d x \in B$ $e v e r y e l e m e n t o f A i s a l s o a n e l e m e n t o f B$ $\Rightarrow A \subset B$ $l e t y \in B$ $\Rightarrow y \in A a n d y \in B$ $\Rightarrow B \subset A$ $A = B$
	Commented by Kunal12588 last updated on 05/Oct/18

	$s i r h e l p f o r t h e n e x t p a r t . i f a n d o n l y i f ? . . . . . . .$
	Answered by arcana last updated on 06/Oct/18

	$si A = B$ $A = A \cup A = A \cup B \lor B = B \cup B = A \cup B$ $\Rightarrow A \cap B = (A \cup B) \cap (A \cup B) = A \cup B$
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