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integers, then we can definitely say that elements like -1, -2,....up to - ∝, all
belong to set S with a membership value zero, while the elements 0, +1,
+2,...to + ∝ belong to set S with membership value one. We can express
this as follows.
S = { 0 / 1.0, +1 / 1.0, +2 / 1.0, +3 / 1.0, ..., + ∝ / 1.0,
-1 / 0.0, -2 / 0.0, -3 / 0.0, ..., - ∝ / 0.0 }
where b in (a / b) form in set S represents the degree of membership of “a”.
Unlike such sets where membership values could be either zero or one, fuzzy
sets represent sets, whose elements can possess degree of membership lying in
the closed interval of [0,1]. As an example, let us consider a set named AGE
which has a range from (0 - 120) years. Now, suppose one assumes the age of
a person by observation, since he (she) does not have a proof of age. We may
classify the person under subset: Young with certain degree of membership,
Old with other membership, Very-Old with a third membership value. For
example, if the age of the person seems to be between (20-22), say, then he
(she) is called young with a degree of membership = 0.9, say, old with a
degree of membership = 0.4, say, and very-old with a degree of membership =
0.2, say. It is to be noted that the sum of these three membership values need
not be one. Now, assume that in the universal set U we have only four ages:
10, 20, 30, 40. Under this circumstance, subsets Young, Old and Very-Old
might take the following form.
Young = { 10 / 0.1, 20 / 0.9, 30/ 0.5 , 40 / 0.3}
Old = { 10 / 0.01, 20 / 0.3, 30 / 0.9, 40 / 0.95 }
Very-Old = { 10 / 0.01, 20 / 0.1, 30 / 0.7, 40/ 0.9}
A question may be raised as to how to get the membership value of the
persons. To compute these from their respective ages, one can use the
membership distribution curves [14], generated intuitively from the
commonsense knowledge (fig. 9.11).
To represent the membership value of an object u in set (subset) A, we
use the notation: µA (u). As an example, the membership value of a person
having age = 80 to belong to subset very-old = 0.7 can be represented as
µ Very-old (age = 80) = 0.7.