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Traditional Bivalent logic uses the boolean operators AND, OR, and NOT to perform the intersect, union and complement operations. These operators work well for bivalent sets and can be essentially defined using the following truth table:
x |
y |
x AND y |
x OR y |
NOT x |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
1 |
1 |
1 |
0 |
0 |
1 |
0 |
1 |
1 |
1 |
1 |
0 |
The truth table above works fine for bivalent logic but fuzzy logic does not have a finite set of possibilities for each input; this makes for an infinitely large truth table. The operators need to be defined as functions for all possible fuzzy values, that is, all real numbers from 0 to 1 inclusive. Fuzzy logic is actually a superset of bivalent logic since it includes the bivalent options (0,1) as well as all reals in between, so a generalized form of these operators will be usefull. The generalized form for these three operators are:
x AND y |
min(x,y) |
x OR y |
max(x,y) |
NOT x |
1 - x |
Using these definitions they can be applied to all of the bivalent combinations above as well as some fuzzy number combinations. The truth table for this can be seen below:
x |
y |
min(x,y) |
max(x,y) |
1 - x |
0 |
0 |
0 |
0 |
1 |
0 |
1 |
0 |
1 |
1 |
1 |
0 |
0 |
1 |
0 |
1 |
1 |
1 |
1 |
0 |
0.2 |
0.5 |
0.2 |
0.5 |
0.8 |
0.7 |
0.2 |
0.2 |
0.7 |
0.3 |
0.6 |
0.6 |
0.6 |
0.6 |
0.4 |
These generalised definitions of the operators work well for fuzzy numbers as well as bivalent sets. The behavior of the new generalized definitions of AND, OR and NOT can be visualised below for all possible inputs of x and y:
The surfaces above show the behaviour of the generalized AND, OR and NOT operators. There are many other possible definitions for the different types of operators but they all share similar properties. Mathematical definitions of the AND operator are called triangular norms or t-norms, this name is derived from the shape of the generalized AND. A t-norm is, by definition, a binary operator with both operand and the result in [0,1], is commutative, associative, has 1 as an identity, and is increasing in each variable. Mathematical definitions of the OR operator have all the same properties of t-norms except that they have 0 as an identity; they are called t-conorms. The NOT operator can be redefined as long as it is a continuous, strictly decreasing function withing [0,1]. Some examples of different t-norms, t-conorms and negations can be seen below:
These are only one example of each different type of operator, many others for each operator exist. The choice of which definition to use will affect the way fuzzy sets are combined and this should be kept in mind when selecting an operators defining function. When an operator definition is selected it should be used consistently so that set combinations remain consistent.