Introduction:-A Fuzzy Set is any set that allows its members to have different degree of membership, called membership function, in the interval [0 , 1].
A fuzzy set A, defined in the universal space X, is a function defined in X which assumes values in the range [0, 1].
A fuzzy set A is written as a set of pairs {x, A(x)} as
A = {{x , A(x)}} , x in the set X
where x is an element of the universal space X, and A(x) is the value of the function A for this element.
The value A(x) is the membership grade of the element x in a fuzzy set A.
Example : Set SMALL in set X consisting of natural numbers ≤ to 12.
Assume: SMALL(1) = 1, SMALL(2) = 1, SMALL(3) = 0.9, SMALL(4) = 0.6,
SMALL(5) = 0.4, SMALL(6) = 0.3, SMALL(7) = 0.2, SMALL(8) = 0.1,
SMALL(u) = 0 for u >= 9.
Set SMALL = {{1, 1 }, {2, 1 }, {3, 0.9}, {4, 0.6}, {5, 0.4}, {6, 0.3}, {7, 0.2}, {8, 0.1}, {9, 0 }, {10, 0 }, {11, 0}, {12, 0}}
A fuzzy set can be defined precisely by associating with each x , its grade of membership in SMALL.
Universal space for fuzzy sets in fuzzy logic was defined only on the integers. Now, the universal space for fuzzy sets and fuzzy relations is defined with three numbers. The first two numbers specify the start and end of the universal space, and the third argument specifies the increment between elements. This gives the user more flexibility in choosing the universal space.
Example : The fuzzy set of numbers, defined in the universal space X = { xi } = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} is presented as SetOption [FuzzySet, UniversalSpace → {1, 12, 1}]
