proof-that-for-two-set-A-and-B-or-give-a-counter-example-A-B-A-B-

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 ∣ \dots \dots \dots . . I

A \cap B \subseteq B \subseteq A \cup B \Rightarrow∣ A \cap B ∣⩽∣ B ∣⩽∣ A \cup B ∣ \dots \dots \dots . . 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 ∣