(Union A n)^c = (Intersect A n)^c

KaliberSun

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Oct 31, 2017
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Hi

English is not my first language. I was wondering if anyone know how to start working this problem. I dont even know where to start. I dont know how to write the correct mathematical symbols on the keyboard so ill post a imgur link.

< link removed by moderator >

How do i show that the complement of the first series amount to the same thing as the complement of the second series? Does anyone have any idea?

Best regards

KaliberSun
 
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I didn't have time to fix your post, earlier.

For readers, here is the appropriate image (properly cropped). Click the thumbnail, to enlarge.

reImgur.jpg
 
What does

\(\displaystyle \displaystyle\bigcup_{j=1}^n A_j\)

mean? Well, it denotes the set of all elements belonging to at least one of the \(\displaystyle A_j\)'s. So

\(\displaystyle \displaystyle\left(\bigcup_{j=1}^n A_j\right)^C\)

means the set of all elements belonging to none of the sets \(\displaystyle A_1, \dots, A_n\). Now look at the other side,

\(\displaystyle \displaystyle\bigcap_{j=1}^nA_j^C.\)

What set does this denote? Try to think about what elements belong to this set. Do you see why this is the same as the previous set we described?

These are De Morgan's laws by the way. You could prove them with induction in this case, but they actually hold even when there are uncountably many sets.
 
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