Logic

[Kival]

Oh, sorry, I did not want to ask if you have a degree, only if you know enough set theory. Okay, ignore my mistakes in english, I hope you can understand the ideas:

Okay let's work with set theory then. a is actually quite trivial but I'll try to give you an idea how to prove it:


Pr(E | H) = Pr(H ∩ E)/Pr(H) (Definition of baysian probability)
=> Pr(H ∩ E) = Pr(E | H)*Pr(H) (just simple transformation)

As Pr(E | H) has a value range in the interval [0;1] (it's a probability function), obviously it follows:

Pr(H ∩ E) <= Pr(E | H)*Pr(H)

Because of Pr(H | E)=Pr(H ∩ E)/Pr(E), the second part of a) follows trivially.

for b) you just need the equations and knew two things:

H → ¬E -> (H ∩ E) = {} and Pr ( {} )=0 ({} being the empty set)

c) Obviously (H ∩ E) and (H^(c) ∩ E) are disjoint sets, so you can use Sigma additivity (H^(c) being the complement of H) . Do you know where to go from there?