If-A-and-B-are-two-sets-and-U-is-a-universal-set-prove-that-A-B-B-A-A-B-

Question Number 1744 by Rasheed Ahmad last updated on 13/Sep/15

I f A a n d B a r e t w o s e t s a n d U i s

a u n i v e r s a l s e t p r o v e t h a t

A \subseteq B \Rightarrow B = A \cup (A^{'} \cap B)

Answered by Rasheed Ahmad last updated on 19/Sep/15

A \subseteq B \Rightarrow B = A \cup (A^{'} \cap B)

\dots \dots \dots . * * * \dots \dots \dots \dots \dots \dots \dots . .

A \subseteq B \Rightarrow A \cup B = B

O r B = A \cup B

= U \cap (A \cup B) [∵ S = U \cap S]

= (A \cup A^{'}) \cap (A \cup B) [∵ U = A \cup A^{'}]

= A \cup (A^{'} \cap B) [∵ (A \cup B) \cap (A \cup C)

= A \cup (B \cap C)]

Answered by arvind last updated on 18/Sep/15

Answered by Rasheed Soomro last updated on 18/Sep/15

A \subseteq B \Rightarrow B = A \cup (A^{'} \cap B)

R H S : A \cup (A^{'} \cap B) = (A \cup A^{'}) \cap (A \cup B)

[D i s t r i b u t i v i t y o f \cup o v e r \cap]

= U \cap (A \cup B) [A \cup A^{'} = U]

= A \cup B [U \cap S = S]

= B [∵ A \subseteq B]

= L H S