Artificial Intelligence - Fuzzy Systems

Question

I have some practice exam questions and answers which are as follows:

However, I am unsure as to why the solution to question 1 (highlighted) begins with 1-... - could somebody please explain this? Do all NOT / ¬ solutions begin with 1-... and if so, why?

Many thanks.

Answer 1

The notation is a bit confusing, because it is mixing fuzzy set theory and fuzzy logic. Here not A probably means the complement (anything but A, sometimes written as A with a bar above the letter A or as A^C) of the fuzzy set A. (Confirm with your tutor later).

The function \mu_A (LaTeX syntax, I do not know, how to enter greek symbols here) is a function which assigns a grade of containment for the set A. I.e. \mu_A(x) = 0.6 means that x is contained in A with a grade of 0.6 (similar but not identical to the probability of x being an element of A). \mu_A(x) = 0 means that x is not an element of A.

So if the grade of containment for x in A is some value v, than a natural definition of the grade of containment in the complement of A (here written as not A) is 1-v (this is also similar to probability theory: if the probability of an element being in some set A is v, then the probability of the element being in A's complement is 1-v).

Thus the containment function \mu_{not A} for the complement can be defined as \mu_{not A} = 1 - \mu_A.

This definition is consistent with union (max) and intersection (min) so that the usual laws of set theory are still correct (like de Morgan's law: the complement of the union of A and B is the intersection of the complements of A and B: not(A u B) = (not A) \intersect (not B)).