proof that for two set A and B (or give a counter example) ∣A∪B∣≥∣A∩B∣


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Question Number 1694 by 123456 last updated on 01/Sep/15

$proof that for two set A and B (or give a counter example)$ $∣ A \cup B ∣⩾∣ A \cap B ∣$
	Answered by Rasheed Soomro last updated on 01/Sep/15

	$In the following A and B are assumed as finite sets .$ $A \cap B \subseteq A \subseteq A \cup B \Rightarrow∣ A \cap B ∣⩽∣ A ∣⩽∣ A \cup B ∣ . . . . . . . . . . . I$ $A \cap B \subseteq B \subseteq A \cup B \Rightarrow∣ A \cap B ∣⩽∣ B ∣⩽∣ A \cup B ∣ . . . . . . . . . . . II$ $From I and II on adding,$ $2 ∣ A \cap B ∣⩽∣ A ∣ + ∣ B ∣⩽ 2 ∣ A \cup B ∣$ $∣ A \cap B ∣⩽ \frac{∣ A ∣ + ∣ B ∣}{2} ⩽∣ A \cup B ∣ [Dividing by 2]$ $∣ A \cup B ∣⩾∣ A \cap B ∣$
	Answered by Rasheed Soomro last updated on 01/Sep/15

	$A \cap B \subseteq A \subseteq A \cup B \Rightarrow∣ A \cap B ∣⩽∣ A ∣⩽∣ A \cup B ∣\Rightarrow∣ A \cup B ∣⩾∣ A \cap B ∣$
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