Some Terminology
A fuzzy rule is a simple IFTHEN rule with a condition and a conclusion.
Sample fuzzy rules for the air conditioner system are as follows:
 1. IF (temperature is warm OR toocold) AND (target is warm), THEN action is heat.
 2. IF (temperature is hot OR toohot) AND (target is warm), THEN action is cool.
 3. IF (temperature is warm) AND (target is warm), THEN action is nochange.
SetTheoretic Operations on Crisp Sets
Crisp sets consist of mishmash of welldefined objects. It is well defined in the sense that an object is applied to a set or not. Here are some of the most important tasks in a crisp set.
Union
The union of two sets A and В is a set that contains elements of both sets A and B. This is indicated by (A U B).
Intersection
The intersection of both sets A and В is a set of amazingly contained elements in both set A and set B. This is indicated by (A n B).
Complement
The complement of set A is the set of all elements in the universal set "E" but not in set A. The complement of set A is denoted by A^{c}
Set Difference
The difference between sets A and В is in A as AВ, but only the set of all elements in В or all other elements in E. This is indicated by (AB).
SetTheoretic Operations for Fuzzy Sets
Given "X" to be universe of discourse, A and В are two fuzzy sets with membership function pA(x) and pB(x); then,
Union
The union of two fuzzy sets A and В is a new fuzzy set AuB. Intersection
Intersection of fuzzy sets A and В is a new fuzzy set A n B.
Equality
Two fuzzy sets A and В are said to be equal, i.e. A=B if and only if pA(x) = pB(x) which ways their membership values must be equal.
Crisp Relations and Compositions on the Same Product Space
A crisp relation R from a set A to a set В assigns to each ordered pair exactly one of the pursuit statements: (i) "a is related to b," or (ii) "a is not related to h."
The Cartesian product A x В is the set of all possible combinations of the items of A and B. For example, when
Relations and Compositions
A crisp relation R from a set A to a set В assigns to each ordered pair exactly one of the pursuit statements: (i) "a is related to b," or (ii) "a is not related to h."
The Cartesian product A x В is the set of all possible combinations of the items of A and B. For example, when
Fuzzy relations map elements of one universe, say U, to those of flipside universe, say V, through the Cartesian product of the two universes. Flowever, the "strength" of the relation between ordered pairs of the two universes is measured with a membership function expressing various "degrees" of strength of the relation on the unit interval [0,1].
As an example, a fuzzy relation "Friend" describes the stratum of friendship between two persons (in unrelatedness to either stuff friend or not stuff friend in classical relation!)
Fuzzy relation "Similarity" U=V = {1,..., 8).
1 
2 
3 
4 
5 
6 
7 
8 

1 
1 
0.5 
0 
0 
0 
0 
0 
0 
2 
0.5 
1 
0.5 
0 
0 
0 
0 
0 
3 
0 
0.5 
1 
0.5 
0 
0 
0 
0 
4 
0 
0 
0.5 
1 
0.5 
0 
0 
0 
5 
0 
0 
0 
0.5 
1 
0.5 
0 
0 
6 
0 
0 
0 
0 
0.5 
1 
0.5 
0 
7 
0 
0 
0 
0 
0 
0.5 
1 
0.5 
8 
0 
0 
0 
0 
0 
0 
0.5 
1 
Let A be a fuzzy set specified on a universe of three discrete temperatures, X = (x„ x_{2}, x_{3}), and В be a fuzzy set specified on a universe of two discrete pressures, Y={y„ y_{2}). Fuzzy set A represents the "ambient" temperature, and fuzzy set В the "nearoptimum" pressure for a unrepealable heat exchanger, and the Cartesian product represents the conditions (temperaturepressure pairs) of the exchanger that are associated with "efficient" operations. For example, let
The fuzzy relation is one kind of fuzzy sets. Therefore, we can wield operations of fuzzy set to the relations (e.g. Union, Intersection, Complement).
A fuzzy relation R is specified on sets A and B, and a flipside fuzzy relation S is specified on sets В and C:
The sonnet S • R=SR of the two relations R and S expresses the relation from A to C.
This sonnet is specified by an inner product. The inner product is similar to an ordinary matrix (dot) product, except that multiplication is replaced by minimum, while summation is replaced by maximum. Thus, this sonnet is specified by the pursuit: