Reasoning Under Uncertainty Questions and Answers

Answer: C

The alpha cut of a fuzzy set contains the elements whose membership is at least alpha. Here the membership is \(\mu_A(x) = x/(x+2)\) and alpha = 0.5, so the condition is

The values of x in the interval [0, 10] that satisfy this are 2, 3, ..., 10. For x = 1 the membership is 1/3, which is less than 0.5, and for x = 0 it is 0, so these are left out. The set is {2, 3, 4, 5, 6, 7, 8, 9, 10}, the third option. The empty set would hold only if no value reached 0.5.

Answer: D

The intersection of two fuzzy sets takes the minimum of the membership values element by element, \(\mu_{A \cap B}(x) = \min(\mu_{A}(x), \mu_{B}(x))\). The pairs are (0.2, 0.1), (0.5, 0.5), (0.6, 0.2), (0.1, 0.7) and (0.9, 0.8), whose minima are 0.1, 0.5, 0.2, 0.1 and 0.8.

So the result is {0.1, 0.5, 0.2, 0.1, 0.8}. Option A takes maxima, which is the union, and options B and C each take a wrong value at one or more positions.

Answer: C

A fuzzy set is called normal when its height h(A) equals 1, that is, at least one element has full membership. The height is the supremum of the membership values, so a set with h(A) = 1 reaches the maximum possible grade.

If h(A) is 0, or less than 0 or less than 1, no element has full membership, so the set is sub-normal. A value below 0 is impossible for a membership function, which lies in the interval from 0 to 1.