Thread: set proof

Could someone please just give me a hint as to how I could prove this:

{(A∩B)’} = 1+P{A∪B} – P{A}– P{B}

Thanks.

Yes, a big THANK YOU, Plato!

Kid

Originally Posted by Plato

Can you (or anyone) explain what rules or laws got you to this? I follow the first step, because it is DeMorgan's Law. But from there I'm lost. I'm looking through all my notes and my texbook, but I'm not seeing the rules that allow you to make those changes.

Originally Posted by ban26ana

Can you (or anyone) explain what rules or laws got you to this? I follow the first step, because it is DeMorgan's Law. But from there I'm lost. I'm looking through all my notes and my texbook, but I'm not seeing the rules that allow you to make those changes.

There's a property (I don't know the name...) that says :



You can see it on a Venn diagram. If you add A and B, you'll add twice


This is why
denotes the complement - see here : Complement (set theory - Wikipedia, the free encyclopedia)
And we have

So :


...

Thanks Moo. I found the theorem in my book that says P(A) = 1 - P(A'). Is that the same one that makes P(A' *intersect* B') = 1 - P(A U B)?

Originally Posted by ban26ana

Thanks Moo. I found the theorem in my book that says P(A) = 1 - P(A'). Is that the same one that makes P(A' *intersect* B') = 1 - P(A U B)?

Yes, because , which is de Morgan's rule.

Thanks again Moo. I get it!