A First Course in FUZZY and NEURAL CONTROL

3.6. TRUTH TABLES FOR FUZZY LOGIC 115

xyx∧yy^0 x∧y^0 qp

00 010 00

01 000 00

10 011 11

11 100 11

The fact thatqandphave the same truth values for every truth value ofxand
yreflects the well-known fact thatpandqare logically equivalent.
For fuzzy logic, it is a remarkable fact thatthree-valued truth tables can
be used for this purpose. Take truth values{ 0 ,u, 1 },where 0 is interpreted as
ìfalse,î 1 is interpreted as ìtrue,î anduas ìother.î For a fuzzy set, the truth
valueucorresponds to the case where the degree of membership, or the truth
value, is greater than 0 but less than 1. The corresponding truth tables are

∨ 0 u 1

0 0 u 1

u uu 1

1 111

∧ 0 u 1

0 000

u 0 uu

1 0 u 1

0

0 1

u u

1 0

These three-valued truth tables can also be used to determine equivalence of
logical expressions of fuzzy sets.

Example 3.10LetA,B:X→[0,1]be fuzzy sets and consider the expressions

p(x)=A(x)∧A^0 (x)

q(x)=(A(x)∧A^0 (x)∧B(x))∨(A(x)∧A^0 (x)∧B^0 (x))

whereA^0 (x)=1−A(x)andB^0 (x)=1−B(x).Alsolet

r(x)=A(x)∧A^0 (x)∧B(x)

s(x)=A(x)∧A^0 (x)∧B^0 (x)

so thatq(x)=r(x)∨s(x). To build a three-valued truth table for a fuzzy set
A:X→[0,1],enter 1 ifA(x)=1, 0 ifA(x)=0,anduotherwise. Likewise
withB:X→[0,1]. Then take all possible values in{ 0 ,u, 1 }forA(x)andB(x)
in thefirst two columns and evaluate the other expressions at those values.

A(x) B(x) A^0 (x) B^0 (x) p(x) r(x) s(x) q(x)

00110000

u 0 u 1 u 0 uu

10010000

0 u 1 u 0000

uuu u uuuu

1 u 0 u 0000

01100000

u 1 u 0 uu 0 u

11000000