a. Consider the following two assertions, where U, V, W, X, Y, and Z are sets of random variables:

(i) U is independent of V given W. 

(ii) X is independent of Y given Z. 

Under what conditions, in terms of subset/superset relationships among U, V, W, X, Y, and Z, can one say that (i) entails (ii)? 

b. We will say that a conditional independence assertion C₁ is strictly weaker than an assertion C₂ if C₂ entails C₁ and C₁ does not entail C₂. For each of the following equations, give the weakest conditional independence assertion required to make it true (if any).

(i)

(ii)

(iii)

(iv)

(v)

(vi)

P(A, C) = P(AB) P(C).