Inclusion properties for random relations under the hypotheses of stochastic independence and non-interactivity
This paper investigates whether random set inclusion is preserved by non-interactivity and by stochastic independence. Let (X-1, x(1)), (X-2, x(2)) be two randomsets on U-1 and U-2, respectively, and let (Y-1, y(1)), (Y-2, y(2)) be two consonant inclusions of theirs. Let (Z(1), z(1)) be the random relation on U-1 x U-2 obtained from (X-1, x(1)) and (X-2, x(2)) under the hypothesis of stochastic independence, and let (Z(2), z(2)) ((Z(3), z(3)), respectively) be the random relation on U-1 x U-2 obtained from (Y-1, y(1)), (Y-2, y(2)) under the hypothesis of non-interactivity ( stochastic independence, respectively). We prove that these hypotheses do not imply that (Z(1), z(1)) subset of (Z(2), z(2)), but imply that (Z(1), z(1)) subset of (Z(3), z(3)).
inclusion relation stochastic independence non-interactivity random set theory