A new intuitionistic fuzzy definiteness norm

In the present paper we introduce a new intuitionistic fuzzy norm over intuitionistic fuzzy pairs and study some of its properties.


Introduction
The Intuitionistic Fuzzy Pair (see [3,6]) is an object with the form a, b , where a, b ∈ [0, 1] and a + b ≤ 1, that is used as an evaluation of some object or process.
The most important geometrical interpretation of the intuitionistic fuzzy pairs is shown on Figure 1.
For the needs of the discussion below, following [1,3], we define the notion of Intuitionistic Fuzzy Tautological Pair by: x is an intuitionistic fuzzy tautological pair if and only if (iff) a ≥ b, while p is a Tautological Pair iff a = 1 and b = 0. Let us have two intuitionistic fuzzy pairs x = a, b and y = c, d . Following [6], we define the relations x < y iff a < c and b > d There are three types of operators over intuitionistic fuzzy pairs, introduced in [6]. The first of them is of modal type and we will introduced only them.
Let as above, x = a, b be an intuitionistic fuzzy pair and let α, β ∈ [0, 1]. Then the modal type of operators defined over x have the forms:

Main results
Let the intuitionistic fuzzy pair x = a, b be given. For it, we define Theorem 1. The definition of δ is correct.
Proof. By assumption x = a, b is an intuitionistic fuzzy pair. We will prove that y = (c + 1 − a − b)a, (d + 1 − a − b)b is also an intuitionistic fuzzy pair for all c, d ∈ [0, 1], and hence also for the particular choice c = 1 − b and d = 1 − a. First, we observe that since x is an intuitionistic fuzzy pair 1 − a − b ≥ 0, and from c ≥ 0 and d ≥ 0, we have that Since we have to show the correctness we need to establish that After rewriting we get The last is equivalent to hence it is always true Therefore, the definition of δ is correct.
The geometrical interpretation of δ(x) is given on Figure 2. Using (1), we check directly the following equalities: for each a ∈ [0, 1]. Proof. Let x be an IFT. We see that the validity of the Theorem follows .
By analogy with the above definitions of intuitionistic fuzzy tautological pair and tautological pair, for a first time we define the notion of Intuitionistic Fuzzy False Pair by: x is an intuitionistic fuzzy false pair iff a < b, while x is a False Pair iff a = 0 and b = 1. Now, we can prove similarly to Theorem 2 the following two assertions  Proof. We check sequentially: Therefore, ¬δ(¬x) = δ(x).

Theorem 6. For each intuitionistic fuzzy pair
Proof. Let x be an intuitionistic fuzzy pair. Then The second equality is checked in the same manner.
Proof. We must find α and β such that: that by condition a = 0 and α = 2 − a − 2b ≤ 1, i.e., α ∈ [0, 1], and from p = 1 − a − b > 0: We must mention that both conditions 2a + b ≤ 1 and a + 2b ≥ 1 imply inequality a ≤ b. Now, we see that The area of the intuitionistic fuzzy pairs over which the operator H α,β can be applied is shown on Figure 3.
The area of the intuitionistic fuzzy pairs over which the operator H * α,β can be applied is shown on Figure 3.  Theorem 11. For an intuitionistic fuzzy pair x such that p > 0, a + 2b ≤ 1 and 2a The proof is similar to this of equality (8). Here, and both conditions a + 2b ≤ 1 and 2a + b ≥ 1 imply inequality a ≥ b.
The area of the intuitionistic fuzzy pairs over which the operator J α,β can be applied is shown on Figure 4.
Theorem 12. For an intuitionistic fuzzy pair x such that a + 2b ≤ 1 and 2a The proof is again similar to this of equality (8). Here, The area of the intuitionistic fuzzy pairs over which the operator J * α,β can be applied is shown on Figure 4.

Conclusion
In future, new properties of the norm δ will be studied. It will be interesting, if it can be used for determining of the areas of positive and negative consonances in the intercriteria analysis (see [2,4,5]).
In this regard, the interested reader can see certain similarities between the newly proposed intuitionistic fuzzy norm and the approach adopted in [7] with the modified level operator N γ . Graphically, that operator cuts off a smaller triangle from the overall intuitionistic fuzzy triangle, by producing a subset of an intuitionistic fuzzy set, which elements satisfy the condition for the ratio between their degrees of membership and their degrees of non-membership, respectively, to be greater or equal to a predefined number γ > 0, i.e., the elements who membership is at least γ times greater than their non-membership.