Let-A-and-B-be-sets-Prove-that-A-B-if-and-only-if-A-B-A-B-

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 \dots \dots (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 ? \dots \dots .

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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