Fuzzy triangular numbers in - Sierpinski triangle and right angle triangle

Sierpinski Triangle or Sierpinski Gasket or Sierpinski Sieve is a fractal and attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. In this paper the number theoretical aspect of Fuzzy Triangular Numbers (FTN) in Sierpinski Triangle and Right Angle Triangle have been established and some arithmetic operations of α – Cut, Centroid of the Triangle, Approximation of the Triangular Fuzzy Numbers are explained.


Fuzzy numbers
The notion of Fuzzy number's as being convex and normal fuzzy set of some referential set was introduced by Zadeh. [1].

Fractal
Fractal was coined by Mandelbrot in his fundamental essay from the Latin fractus meaning broken, to describe the objects that were too irregular to fit into a traditional geometrical setting [2]. Many fractals have same degree of self-similarity-it made up parts that resemble the whole in some way. Sometimes, the resemblance may be weaker than strict geometrical similarity for example the similarity may be approximate or statistical. Sierpinski Triangle, Cantor Set and Von Koch Curve, Menger sponge, Dragon Curve, Julia Set, Mandelbrot Set are the examples of Fractals. [2,9] Sierpinski triangle is a fractal described in 1915 by Waclaw Sierpinski. It is a self-similar structure that occurs at different levels of iterations, or magnifications. The Sierpinski Triangle has all the properties of a Fractal.

Sierpinski triangle
For triangular area, with each iteration, the side of the inside triangle reduces by a factor of 2. The numbers of these little triangles, on the other had increases not by 4 but by factor of 3. The dimension of self-similar object is then (log3/ log 2) = 1.58 approximately [3]. In this paper in Section 2, Definitions are explained. In Section 3, the operations of Fuzzy Triangular Numbers are discussed using Sierpinski triangle with side 1unit, 2unit, 3unit and 12 unit and Right angle triangle with side 6,8,10 units. [ [5], they all discussed with randomly chosen Triangular fuzzy numbers. But in this paper Triangular Fuzzy Numbers are chosen from the selfsimilarity Fractal set Sierpinski triangle and Right angle Triangle explained the basic operations of addition, subtraction ,Approximation of Two fuzzy Triangular Numbers multiplication and division including α -Cut operations with graphical representation.

Preliminaries
2.1 Fuzzy set [6]. Any set which allows its members to have membership of different grades in the interval [0, 1] is a Fuzzy Set.

Support of fuzzy set
The support of a fuzzy set F is a crisp set of all points in the Universe of Discourse U (range of all possible values for an input to a Fuzzy system) such that the membership function of F is non-zero [6].

Fuzzy number [1]
A fuzzy set A on R must possess at least the following three properties to qualify as a fuzzy number,  A must be a normal fuzzy set;  A α must be closed interval for every α [0,1]  The support of A , A 0+ must be bounded

Triangular fuzzy numbers (TFN) [4]
Let a, b, and c be real numbers with a < b< c. Then the Triangular Fuzzy Number (TFN) A= (a, b, c) is a Fuzzy Number with membership function: Operation of triangular fuzzy number using function principle [1] The following are the four operations that can be performed on triangular fuzzy numbers:

Centroid of the triangle [4]
The coordinates (X, Y) of the COG of the triangle forming the graph of the TFN (a, b, c) are calculated by the formulas  Figure.1 represents of Fuzzy Triangular Numbers of Equilateral Triangle of side 1 unit

Operations on fuzzy triangular numbers 3.1 Equilateral triangle
Solving equation (1) and (2), based on the definition 2.8 the coordinates of Centroid of the triangle G are (6, 1/3) which is equal to x= .
Case II [7] When any and is partial negative and other is positive fuzzy number, the product of can not obtain according to the equation (6). The interval of α will be divided into two parts, according Case III When A ̃ is negative and B ̃ is positive Fuzzy number. Then the multiplication of Ã and B can be found as equation (6).

Case IV
When A ̃ and B ̃ both are negative fuzzy number. Then the multiplication of A ̃ and B ̃ can also be found as equation (6).The equation (6) and (7) are known as analytical method for fuzzy arithmetic operation.
In actual product the lines connecting the ends points are parabolic and in standard approximation lines connecting the ends points are triangular form.