Probability of unions/intersections

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The discussion centers on the validity of a probability formula involving unions and intersections of sets. The formula suggests that the probability of the union of intersections of sets is equal to the probability of the union of those sets minus the intersection of the last two sets. While one participant confirms the formula holds for the case where k equals m+1, they express uncertainty about its general validity for k greater than m. They propose using mathematical induction to prove the formula, having established a base case but struggling with the inductive step. The conversation highlights the complexity of the algebra involved and the potential for confusion in proving the formula's correctness.
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Is it true that

Pr( ∪_(n from m to k) ((A_n) ∩ ((A_(n+1))^c)) )
= Pr( ∪_(n from m to k) (A_n) ) - Pr( (A_k) ∩ (A_k+1) )

where A_1, A_2, ... is any sequence of sets.



Well, for the (k=m+1) case I am convinced since I can see they are equal after expanding both sides out, so for example I can see that
Pr((A∩(B^c))∪(B∩(C^c))) = Pr(A∪B) - Pr(B∩C)

but I can't manage to do the same for the (k>m) case in general, so overall I'm not convinced.
 
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I'm sorry I can't make much sense out of the formula, but assuming its correct, try induction. Assume the proposition is true for k <= m, and add one more term to it and use the truth of the m previous propositions to prove it for (m+1). Will require some grouping and basic properties of sets and cardinalities under union and intersections.
 
That (induction) is exactly what I've been attempting to use to convince myself that it is true. I've got the base step (k=m+1) which was (for me) expand-able to see that both sides are equal.
However I couldn't get through the inductive step. Perhaps it is false then? Or it could also just mean that I got totally lost within the messy algebra?
 
You have a "special symbol" right at the beginning of that formula that will not show up on my (or Maverick280857's) browser.
 
Pr( ∪_(n from m to k) ((A_n) ∩ ((A_(n+1))^c)) )
= Pr( ∪_(n from m to k) (A_n) ) - Pr( (A_k) ∩ (A_k+1) )



or



the probability of [ the union (where n goes from m to k) of [ A_n intersect (A_(n+1) compliment) ] ]

is equal to

the probability of [ the union (where n goes from m to k) of A_n ]
minus
the probability of [ A_k intersect A_(k+1) ]
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
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