ABSTRACT
We know that the classical congruence is defined on integers. The module of congruence is integer too. For an integer m, there are m classes of congruence that they are partitioned integers into m parts. These classes will be determined by dividing the integers on the remaining m. these residues are 0, 1, 2, . . . ,m− 1. When any real number is assigned to the integer m, the rest is a real number in the interval [0,m− 1]. Therefore we can extend congruence on integers to the real numbers so that any arbitrary real number x belongs to a class of congruence module m, with a membership degree. In other words we can say:
Or:
Means that a real number x − a is a fuzzy congruence zero of module m, with membership degree y(x) that function y(x) is defined as following:
Therefore for every real number x, membership function of fuzzy congruence module m is defined as follows:
1.1 Fuzzy congruence of two triangle fuzzy numbers
Now we assume that a˜ and b˜ are two different triangle fuzzy numbers. We can determine congruence of
this two triangle fuzzy numbers to module m, in other words we can specify:
First transform these two fuzzy numbers with an order function into two corresponding real numbers. An order function is defined as following:
The reason of function definition is that two triangle fuzzy numbers may have overlap. By the way, two fuzzy numbers are less likely overlap and therefore the result of calculations can be achieved easier and more accurate. It is worth noting that one of the characteristics of order function is that it is linear function.
