Question 1. Suppose X = {-3, -2, -1, 0, 1, 2, 3}. Let fuzzy subsets A and B of X be defined by their membership vectors A = (0.0, 0.3, 0.8, 1.0, 0.8, 0.3, 0.0) and B = (1.0, 0.9, 0.7, 0.5, 0.2, 0.0, 0.0)

A. Using Zadeh's original definitions, compute

A^c

A ∪ B

A ∩ B

B. What is A ∪ B if A ∪ B(x) = 1 Λ (A(x) + B(x))?

Question 2. Show that Demorgan's laws hold for the standard fuzzy set theory definitions, that is, that (A ∪ B)^c = A^c ∩ B^c  and (A∩B)^c = A^c ∪ B^c.

Question 3. For the fuzzy sets defined in Question 1(A), generate all of the nonzero values of (A+B) and Max(A,B).

Question 4. Let X = {1,  2,  3}; Y = {a, b}; Z = {w, x, y, z}; and

Let A = 1.0/1 + 0.4/2 + 0.1/3

B = 0.2/a + 0.8/b

C = 0.0/w + 0.4/x + 0.8/y + 1.0/z

Consider the rule: IF U is A THEN W is C

A. Translate this rule using Zadeh's original translation formula (i.e., the Lukasiewicz implication).

B. Show the result of inference if the input is

i. U is A;

ii. U is NOT A;

iii. U is A' where A' = 0.6/1 + 1.0/2 + 0.0/3.

C. Translate the following rule using Correlation-Min encoding:

If U₁ is A and U₂ is B THEN W is C

Question 5. Let X = {a, b, c, d, e, f}; Y = {1, 2, 3, 4}; Z = {s, t}; and

let A = 0.0/a + 0.0/b + 1.0/c + 1.0/d+ 0.0/e + 1.0/f

B = 1.0/1 + 0.7/2 + 0.2/3 + 0.0/4

C = .9s + 0.5/t

Consider the rule: IF V is A THEN W is B

A. Translate this rule using Zadeh's original translation formula (i.e., the Lukasiewicz implication).

B. Show the result of the compositional rule of inference (for the translation in part A) when the input is "V is A."

C. Based on the result of part B, state and prove a theorem about the "firing" of rules that have a crisp antecedent.