Inference

In the inference method we utilized knowledge to perform deductive reasoning. This is we wish to conclude or infer a conclusion, specified a body of facts and knowledge. There are many forms of this method documented in the literature, however the one we will illustrate now relates to our formal knowledge of geometric and geometry shapes. In the identification of triangle, assume A, B and C be the inner angles of a triangle, in the order A ≥ B ≥ C ≥ 0, and assume U, be the universe of triangles that is;

                                  U = {(A, B, C) | A ≥ B ≥ C ≥ 0; A + B + C = 180^o}...................Eqn(2)

We explain a number of geometric shapes that we wish to be capable to identify for any collection of angles fulfilling the constraint given in eq...2. For this reason we will define the given five types of triangles as:

• I as approximate isosceles triangle
• R as Approximate right triangle
• IR as Approximate isosceles and right triangle
• E as Approximate Equilateral triangle
• T as Other triangles

We can conclude membership values for all these triangle prototypes through the method of inference, since we posses knowledge about geometry that helps us to make membership assignments. Then we shall list this knowledge now to develop an algorithm to assist us in making these membership assignments for some collection of angles meeting the constraints of Eq...2.

For approximate isosceles triangle we have the given algorithm for the membership, again for the condition of

                                                            A ≥ B ≥ C ≥ 0 and A + B + C = 180^o;

                                                 μ1 (A, B, C) = 1 - (1/160^o) min (A - B, B - C).............Eqn(3)

So, for illustration if A = B or B = C, the membership value in the approximate isosceles triangle is μ_I = 1; if A = 1200, B = 60^o, and C = 0, then μ_I = 0. For fuzzy right triangle, μR = 1, or while A = 180^o this membership vanishes that is μR = 0. For the case of isosceles and right angle triangle, we can determine this membership function by considering the logical intersection of the isosceles and right angle triangle membership functions, or else

                                                                       IR = I ∩ R...................Eqn(4)

That results in:

 μ_IR (A,B,C) =  min [μ_I (A,B,C) μ_r (A,B,C)

                                                              1 - max[(1/160⁰)min (A-B)(B-C),(1/90⁰)¦A-90¦]..................Eqn(5)

For case of the fuzzy Equilateral triangle, the membership function is specified by:

                                                                     μ_B (A,B,C)  = 1 - [(1/180⁰)(A-C)]............................Eqn(6)

For illustration when A = B = C, the membership value is μ_E (A, B, C) = 1; while A = 180^o,

Introduction to the membership value vanishes, or μ_E= 0. Finally, for the set of "all other triangles" we basically invoke the complement of the logical union of the three previous cases.