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 May 15th, 2017, 05:15 AM #1 Senior Member   Joined: Nov 2015 From: hyderabad Posts: 211 Thanks: 2 Property verification Hello Please let me know if the below property is always true or does it need any additional conditions!!! Let X be a set. For $\displaystyle A \subset X$, let $A^c = X-A$. The correct statement for $\displaystyle A ,B \subset X$ is: A) $A-B=B^c -A^c$, always B) If $A-B=B^c -A^c$ then $\displaystyle A \subset B$ or $\displaystyle B \subset A$. C) If $A-B=B^c -A^c$ then $\displaystyle A \cap B = \emptyset$ D) If $A-B=B^c -A^c$ then $A=X$ or $B=X$ I think its always true because I checked the other 3 options. Please let me know if I'm wrong
 May 15th, 2017, 02:48 PM #2 Member   Joined: Jan 2016 From: Athens, OH Posts: 69 Thanks: 37 You're right. For any subsets S and T, $S-T=S\cap T^c$. So $$B^c-A^c=B^c\cap(A^c)^c=B^c\cap A=A\cap B^c=A-B$$
May 15th, 2017, 06:53 PM   #3
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Quote:
 Originally Posted by johng40 You're right. For any subsets S and T, $S-T=S\cap T^c$. So $$B^c-A^c=B^c\cap(A^c)^c=B^c\cap A=A\cap B^c=A-B$$
Thanks for the confirmation

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