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May 15th, 2017, 04:15 AM   #1
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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
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May 15th, 2017, 01:48 PM   #2
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You're right. For any subsets S and T, $S-T=S\cap T^c$. So
$$B^c-A^c=B^c\cap(A^c)^c=B^c\cap A=A\cap B^c=A-B$$
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May 15th, 2017, 05:53 PM   #3
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Quote:
Originally Posted by johng40 View Post
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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