Chapter X: Introduction to Fuzzy Set Theory Uncertainty is universal ...

( A ∪ B)(x) = 1 ∧ (A(x) + B(x))
( A ∩ B)(x) = 0 ∨ (A(x) + B(x) – 1)
A c (x) = 1 – A(x)
A. Show that Intersection and Complement satisfy the Law of Contradiction.
B. Is it true that Intersection is idempotent: A ∩ A = A ? (prove or give a counterexample)
3. Let A(x) =
⎧ 0
⎪
⎨x + 3
⎪
⎩-
x -1
if x < - 3 or x > -1
if - 3 ≤ x ≤ - 2
if - 2 < x ≤ -1
and
B(x) =
⎧ 0
⎪
⎨ x -1
⎪
⎩-
x + 3
if x < 1or x > 3
if 1 ≤ x ≤ 2
if 2 < x ≤ 3
A. Compute 0.7 A.
B. Using the standard definitions, sketch a picture of B c .
C. Now, let C(x) =
⎧ 0
⎪
⎨ x - 2
⎪
⎩-
x + 4
if x < 2 or x > 4
if 2 ≤ x ≤ 3
if 3 < x ≤ 4
Compute
B ∩ C
4. Briefly discuss this statement: There is no need for fuzzy set theory because all uncertainty can be
modeled by probability theory. (1 or 2 short paragraphs only)
5. Let A = .5/a + .4/b + .7/c + .8/d + 1/e. List all non-empty α-cuts of A.