Investigation of Order among Some Known T-norms
Shohel Babu^{1}, Fatema Tuj Johora^{1}, Abdul Alim^{2}
^{1}Mathematics, IUBAT-International University of Business Agriculture and Technology, Dhaka, Bangladesh
^{2}Mathematics, BGMEA University of Fashion and Technology, Dhaka, Bangladesh
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To cite this article:
Shohel Babu, Fatema Tuj Johora, Abdul Alim. Investigation of Order among Some Known T-norms.American Journal of Applied Mathematics.Vol.3, No. 5, 2015, pp. 229-232. doi: 10.11648/j.ajam.20150305.14
Abstract: In this paper, order among some known T-norms is investigated. Firstly, the T-norm which is the strongest or greatest and the T-norm which is the weakest is observed. Comparing two T-norms we establish the relation which is strong or weak. In addition, for parametric T-norms after changing the interval of their parameter a relation has established which is strong or weak. Finally, compared has done among three or more T-norms.
Keywords: T-norm Algebraic Product , T-norm Min T-norm Drastic Product T-norm Franks Product , T-norm Einstein Product , T-norm Hamacher Product , T-norm Dubois & Prade Product
1. Introduction
T-norms or triangular norms are generalization of the classical triangular inequality according to K. Menger in 1942. In 1960, B. Schweizer and A. Sklar after revision of this work redefined the concept of triangular norm as an associative and commutative binary operation. They play a fundamental role in probabilistic metric spaces, probabilistic norms and scalar products, multiple-valued logic, fuzzy sets theory.
Definition:
A T-norm is a function: [0, 1] × [0, 1] → [0, 1] which satisfies the following properties:
[Monotonicity:
[Commutativity:
[Associativity:
[Boundary condition:
2. Ordering of T-norms
2.1. Proposition: Among all T-norm "Min" is the Strongest
Proof: Suppose T is any T-norm.
I;
[by boundary condition]
= [by boundary condition]
So
Again [by commutativity]
[by monotonicity]
=
So
Consequently,is a lower bound of .
Again is the greatest lower bound(glb) of .
But I is in chain, so .
Hence "min" is the strongest or greatest t-norm.
2.2. Proposition: Among all T-norm "Drastic Product" is the Weakest
Proof: Suppose T is any T-norm.
I; there are two cases:
Case1: Without loss of generality suppose =1, then
and [by boundary condition].
Consequently =1.
Case2: When 1 and 1, then
But in I; clearly, 0
Therefore.
Hence among all T-norm is the weakest.
2.3. Proposition: Letkϵ]0,1[, Frank’S Product
And Algebraic Product
Then
Proof: Let and
Then and which means that is monotonically decreasing.
If we have then by mean value theorem we get
because
i.e.
Now let
Here
Which means that is monotonically decreasing and,
Furthermore,
i.e
This means that
Again, we clam that
And
Hence
2.4. Proposition: Letkϵ]1, ∞[, Frank’s Product (x,y)k=logk (1+(k^{x}-1)(k^{y}-1)/k-1)
and Algebraic Product
Then
Proof: Let and
Then and which means that is monotonically decreasing.
Now we have
and
because is monotonically decreasing and
i.e.
(1)
Now let
We see that
According to (1) and is monotonically decreasing.
Then,
i.e.
(2)
From (2) we get
This means that
Moreover
And
Hence
2.5. Proposition: Letkϵ]1, ∞[, Frank’s Product
And Boundary Product
Then
Proof: We will distinguish two cases:
Case1: If then
Now for we get
Which is true, because and
Case2: If then
For
This is obvious.
Hence
2.6. Proposition: Letkϵ[0,1], Duboi’s & Prade Product
Then
Proof: We will distinguish three cases, according to the maximum value of .
Caes1: If , then
=
Case2: If , then
Case3: If , then
Hence we can say from three cases
2.7. Proposition: Let Einstein Product TE:I×I→I
Then
Proof: Now we will split the proof into two cases:
Case1: If , then
Which is true, because
Case2: If then
(i)If , then
In this case we get,
Since
(i)Otherwise
This is true
From above we may conclude that
2.8. Proposition: Let Hamacher Product TH:I×I→I
Then
Proof: As above we will distinguish two cases:
Case1: In this case, it will prove that , then
This is obvious, since
Case2: Now we prove the relation between .
We may see that if then
Otherwise, if , and consider , then
This is true because
Since
Hence we may conclude that
3. Conclusion
Form above discussions, it conclude that T-norm Min is the strongest and T-norm Drastic product is the weakest T-norms. Also, T-norm Algebraic product is stronger than T-norm Hamacher product and Boundery Product. Similarly, T-norm Dubois & Prade product is stronger than T-norm Algebraic product. Since, T-norms are using for taking suitable decision from multi-valued logic. So, the ordering of T-norms, that’s means, this paper will help to take correct decision using sufficient T-norms according to their order.
References