A quite different possibility to define a fuzzy/probabilistic subset relation:
Assume the sets A and B are events (sets of possible disjoint outcomes). Then A⊆B iff P(B∣A)=1. This suggests that a probabilistic/partial/fuzzy “degree of subsethood” of A in B is simply equal to the probability P(B∣A).
This value is 1 if A is completely inside B, reducing to conventional crisp subsethood, and 0 if A is completely outside B. It is 0.5 if A is “halfway” inside B. Which seem pretty intuitive properties for fuzzy subsethood.
Additionally, the value itself has a simple probabilistic interpretation—the probability that an outcome is in B given that it is in A.
A quite different possibility to define a fuzzy/probabilistic subset relation:
Assume the sets A and B are events (sets of possible disjoint outcomes). Then A⊆B iff P(B∣A)=1. This suggests that a probabilistic/partial/fuzzy “degree of subsethood” of A in B is simply equal to the probability P(B∣A).
This value is 1 if A is completely inside B, reducing to conventional crisp subsethood, and 0 if A is completely outside B. It is 0.5 if A is “halfway” inside B. Which seem pretty intuitive properties for fuzzy subsethood.
Additionally, the value itself has a simple probabilistic interpretation—the probability that an outcome is in B given that it is in A.