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May 15th, 2017, 04:15 AM  #1 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 230 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 = XA$. The correct statement for $\displaystyle A ,B \subset X $ is: A) $AB=B^c A^c$, always B) If $AB=B^c A^c$ then $\displaystyle A \subset B $ or $\displaystyle B \subset A $. C) If $AB=B^c A^c$ then $\displaystyle A \cap B = \emptyset $ D) If $AB=B^c A^c$ 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, 01:48 PM  #2 
Member Joined: Jan 2016 From: Athens, OH Posts: 87 Thanks: 45 
You're right. For any subsets S and T, $ST=S\cap T^c$. So $$B^cA^c=B^c\cap(A^c)^c=B^c\cap A=A\cap B^c=AB$$ 
May 15th, 2017, 05:53 PM  #3 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 230 Thanks: 2  

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