1

We know from the theory the following properties:

  1. 0≤P(A)≤1 for every A
  2. P(Ω)=1
  3. If A1, A2,...An are mutually exclusive then P(A1 ∪ A2 ∪ ...An)= P(A1)+P(A2)+...+P(An)

In the exercise, it is given that Ω= {α,β} and ΣΩ={ Ø , Ω, {α}, {β} }

It is asked:

i) to find a set function Q: ΣΩ → R that satisfies 1 and 2 but not 3

ii) to find a set function Q: ΣΩ → R that satisfies 1 and 3 but not 2

I tried to start it by saying that: P({a})= q and P({b})=1-q , but i find it impossible not to meet one of these properties while meeting the other two.

Andrés E. Caicedo
  • 79,201
therealbishop
  • 13

1 Answers1

0

$i)$ $Q(\emptyset)=0, Q(\Omega)=1, Q(\{\alpha\})=0.9, Q(\{\beta\})=0.9.$

$ii)$ $Q(\emptyset)=0, Q(\Omega)=0, Q(\{\alpha\})=0, Q(\{\beta\})=0.$

Riccardo
  • 1,304
  • How is this true? Given that the powerset only contains two elements ({α} and {β}), shouldn't it be: Q({α})=0.9 Q({β})=1-Q({α})=1-0.9=0.1 ? – therealbishop May 03 '20 at 20:18
  • The powerset contains four elements and $Q$ does not have to satisfy your additional property. Notice that $you$ assign values. – Riccardo May 03 '20 at 20:27