On Interval-Valued Fuzzy on Ideal Sets

Fuzzy sets, formalized by Zadeh in 1965, generalizes the classical idea of sets. The idea itself was generalized in 1975 when Zadeh introduced the interval-valued fuzzy sets. In this paper, we generalize further the above concepts by introducing interval-valued fuzzy on ideal sets, where an ideal is a nonempty collection of sets with a property describing the notion of smallness. We develop its basic concepts and properties and consider how one can create mappings of intervalvalued fuzzy on ideal sets from mappings of ordinary sets. We then consider topology and continuity with respect to these sets. 2010 Mathematics Subject Classifications: 03E72, 62B86, 94D05


Introduction
In classical set theory, an element either belongs or does not belong to a given set.That is, the membership of elements to a given set is assessed in binary terms.Thus, one may associate a set A on a universal set U to the characteristic function of A with values 0 or 1.However, there are informations that cannot be precisely assessed as belonging to or not to a given set, like the set of young people in a group.To address this problem, in 1965, Zadeh [9] and Klaua [4] introduced fuzzy sets, where elements have degrees of membership, not just 0 or 1. Formally defined, a fuzzy set is a mapping from U into the unit interval [0, 1].In our example, for a not so young member of the group, a degree of membership equal to 0.2 can be assigned.
In 1978, Zadeh used his theory of fuzzy sets and fuzzy logic to introduce possibility theory [11].The theory uses a possibility distribution which should not be confused with a probability distribution.Both are fuzzy sets but the sum of the values of a possibility distribution need not be 1 while it should be 1 in a probability distribution.For instance, we may assign a value of 0.4 for the possibility that tomorrow there will be rain and a value of 0.7 for the possibility that tomorrow will be sunny.The sum of these two possibilities is already greater than 1, but our assignment may represent best the information that we know about what will be the weather for tomorrow.Informations like these, with a lot of uncertainties, are not suited to be expressed using a probability distribution.Now, consider the possibility that tomorrow there will be rain and at the same time it will be sunny.This case is not impossible as it happens rarely in the Philippines.But its possibility should be far less than any of the two separate possibilities.We could not just give it a value equal to the minimum of the two separate possibilities.Expressing information like this motivated the introduction of fuzzy on ideal sets in [6] by Mernilo and Caga-anan.An ideal here is a nonempty collection of subsets of a set X, denoted by I(X), that satisfies: i.A ∈ I(X) and B ⊆ A implies B ∈ I(X); and ii.A ∈ I(X) and B ∈ I(X) implies A ∪ B ∈ I(X).
The first property is the reason why an ideal is said to be a collection of sets that are considered small.Ideal spaces were first studied by Kuratowski [5] and Vaidyanathaswamy [8].
Formally, given a nonempty set X and an ideal I(X) on X, a fuzzy on ideal set is a mapping µ : I(X) → [0, 1] such that: i. µ(∅) = 0; and ii. for nonempty sets A, B ∈ I(X), with A ⊆ B, we have µ(B) ≤ µ(A).
The set of all such µ is denoted by I I(X) .Observe that the reverse inequality µ(B) ≤ µ(A) encapsulates the preceding idea that the possibility that tomorrow there will be rain and at the same time it will be sunny should not just be equal to the minimum of the separate possibilies as it could be far less.It is also important to note that the preceding definition does not define a measure.For A ⊆ B, a measure m should have m(A) ≤ m(B), not the reverse inequality, as in our definition.Moreover, any fuzzy set α defined on a set X can be embedded as an element of I P(X) by associating it with the fuzzy on ideal set µ α defined by where P(X) is the powerset of X−the largest ideal of X.Thus, fuzzy on ideal sets generalize fuzzy sets.With regard to uncertainty, there are cases that even the assigning of degrees of membership on a fuzzy set or fuzzy on ideal set has its own uncertainties.In these cases, it is better to give the degree of membership as an interval rather than as a single number.For instance, when one is estimating the age of a person, one has a better chance of capturing the real age by giving a possible range of the age rather than estimating it with a single number.This way, one captures the imprecision better.This lead to the introduction of interval-valued fuzzy sets in 1975 by Zadeh [10].In that same year, it was also considered by Grattan-Guiness [2], Jahn [3] and Sambuc [7].
In this study, we introduce and develop the interval-valued fuzzy on ideal sets.This concept generalizes the above discussed fuzzy sets, fuzzy on ideal sets, and interval-valued fuzzy sets.We formally define it in the next section.

Basic concepts and properties
Let us first introduce some useful notations.We denote by I the set of all closed subintervals of [0,1].For α ∈ I , let α − be the left endpoint of α and α + be the right endpoint of α, so that α be closed subintervals of I .We use the inequality notation "≤ I ", say Let A be an index set and α i ∈ I , for each i ∈ A. We define the supremum of α i by sup ] and the infimum of α i by inf We define formally an interval-valued fuzzy on ideal set as follows.
Definition 1.Let X be a nonempty set and I(X) be an ideal on X.An interval-valued fuzzy on ideal set (briefly an IVFI set) is a mapping ι : I(X) → I that satisfies the following: We denote the set of all such ι by I I(X) .Remark 1.In a similar way that fuzzy on ideal sets generalize fuzzy sets, as discussed above, IVFI sets generalize interval-valued fuzzy sets.
Example 1.Let X be a nonempty set and π : X → I be an interval-valued fuzzy set.We can define an IVFI set π : P(X) → I by This is similar to the guaranteed possibility given in [1].
We call an IVFI set ι : I(X) → I with the property that for all nonempty set A ∈ I(X), ι(A) = inf x∈A ι({x}), a guaranteed possibility IVFI set.Remark 2. Let X be a nonempty set and I(X) be an ideal on X.We denote by 0 I(X) the IVFI set 0 I(X) : I(X) → {[0, 0]} and by 1 I(X) the IVFI set Next, we define some relational operators between IVFI sets.
Definition 2. Let X be a nonempty set and I(X) be an ideal on X.Let ι, τ ∈ I I(X) .We say (i) ι is a subset of τ , denoted by ι τ , if ι(A) ≤ I τ (A), for all A ∈ I(X); and (ii) ι is equal to τ , denoted by ι = τ , if ι(A) = τ (A), for all A ∈ I(X).
Remark 3. To avoid confusion, we summarize first our "inequality" notations.
i.The symbol "≤" for the usual inequality with the real numbers.
ii.The symbol "≤ I " for the inequality with intervals.
iii.The symbol " " to denote the subset relation with IVFI sets.
Next, we define complement, union, and intersection of IVFI sets.We then prove that the resulting mappings are also IVFI sets, showing that our definitions are well-defined.Definition 3. Let X be a nonempty set and I(X) be an ideal on X.Let ι ∈ I I(X) .The complement of ι, denoted by ι c , is defined by, ι c (∅) = [0, 0] and for every nonempty set A ∈ I(X), Let X be a nonempty set and I(X) be an ideal on X.If ι ∈ I I(X) , then the complement of ι is an IVFI set.
Proof.Let ι ∈ I I(X) .Let A ∈ I(X), x ∈ A and ι({x} and indeed we have a closed interval.We need to show that the reverse inequality of an IVFI set holds.
Therefore, ι c is an IVFI set.
Remark 4. For a singleton set A = {x} ∈ I(X), the preceding definition coincides with the definition of the complement of an interval-valued fuzzy set.Definition 4. Let X be a nonempty set and I(X) be an ideal on X.Let ι, τ ∈ I I(X) .The union and intersection of ι and τ , denoted by ι ∨ τ and ι ∧ τ , respectively, are given by for all A ∈ I(X), respectively.
In general, the union and intersection of a collection of IVFI sets { ι j : j ∈ J}, denoted by j∈J ι j and j∈J ι j , are given by for all A ∈ I(X), respectively.One can easily check that the arbitrary union or intersection of IVFI sets is an IVFI set.
The following are some properties of the operations on IVFI sets.

iv. (Distributivity
Summarizing the two cases, we have We also consider two cases.First, if τ ({x}) ≤ I ι({x}), then Combining the two cases, we have Hence, ( ι∨ τ ) c = ι c ∧ τ c .Using the same argument, we can also show that ( ι∧ τ ) c = ι c ∨ τ c , and our proof is complete.
The next theorem states some interesting properties of the complement of IVFI sets.Theorem 2. Let X be a nonempty set and I(X) be an ideal on X.Let ι, τ ∈ I I(X) .Then, i. ι ( ι c ) c ; ii.ι = ( ι c ) c if and only if ι is a guaranteed possibility IVFI set; and iii.if ι τ , then τ c ι c .

Mappings
Let X and Y be nonempty sets and f : X → Y be a mapping.Moreover, let I(X) and I(Y ) be ideals on X and Y , respectively.We define the image and pre-image of the ideals under f by f (I(X)) = {f (A) : where f (A) and f −1 (B) is the usual image and preimage of A ⊆ X and B ⊆ Y , respectively.The next theorem is important because it shows that these image and preimage of ideals are also ideals.The proof can be found in [6].

Theorem 3 ([6]
).Let X and Y be nonempty sets and let f : X → Y be a mapping.If I(X) and I(Y ) are ideals on X and Y , respectively, then f (I(X)) and f −1 (I(Y )) are ideals on Y and X, respectively.
Given a mapping of two ordinary sets, we define the image and pre-image of IVFI sets.We then prove that these image and pre-image are also IVFI sets, showing that they are well-defined.Definition 5. Let X and Y be nonempty sets and f : X → Y be a mapping.Moreover, let I(X) and I(Y ) be ideals on X and Y , respectively.
where S = {A ∈ I(X) : ii.If τ ∈ I I(Y ) , then the pre-image of τ under f , denoted by where Remark 5. Let X and Y be nonempty sets and f : X → Y be a mapping.Let ι and τ be IVFI sets in I I(X) and I I(Y ) , respectively.Then, we have f [ ι](∅) = [0, 0] and Theorem 4. Let X and Y be nonempty sets and f : X → Y be a mapping.Let ι and τ be IVFI sets defined on the ideals I(X) and I(Y ), respectively.Then, f [ ι] and f −1 [ τ ] are IVFI sets defined on the ideals f (I(X)) and f −1 (I(Y )), respectively.
Proof.We first show that f [ ι] is an IVFI set defined on the ideal f (I(X)).Let ).Therefore, with Remark 5 and Theorem 3, f [ ι] is an IVFI set defined on the ideal f (I(X)).Next, to show that f . Therefore, with Remark 5 and Theorem 3, f −1 [ τ ] is an IVFI set defined on the ideal f −1 (I(Y )).
We can extend our result to composition of mappings.The following corollaries are immediate consequences of Theorem 3 and Theorem 5.
Corollary 1.Let X, Y , and Z be nonempty sets and f : X → Y and g : Y → Z be mappings.Let g •f : X → Z be a composition map.If I(X) and I(Z) are ideals on X and Z, respectively, then, (g • f )(I(X)) = g(f (I(X))) and (g • f ) −1 (I(Z)) = f −1 (g −1 (I(Z))) are ideals on Z and X, respectively.
Corollary 2. Let X, Y , and Z be nonempty sets and f : X → Y and g : Y → Z be mappings.Let g • f : X → Z be a composition map and, I(X) and I(Z) are ideals on X and Z, respectively.If ι ∈ I I(X) and η ∈ I I(Z) , then The next theorem state some properties of the defined mappings of IVFI sets.We start with the following needed proposition.Proposition 2. Let X and Y be nonempty sets and, I(X) and I(Y ) be ideals in X and Y , respectively.Let f : X → Y be a mapping.Then, i.
Suppose that f is one-to-one.Let A ∈ f −1 (f (I(X))).Then there exists Theorem 5. Let X and Y be nonempty sets and, I(X) and I(Y ) be ideals in X and Y , respectively.Let f : X → Y be a mapping.If ι, τ ∈ I I(X) and ω, η ii.Let ∅ = B ∈ f (I(X)) and S = {A ∈ I(X) : where S = {A ∈ I(X) : Hence, where A B = {x : {x} ∈ S y , y ∈ B}.Thus, iv.Let B ∈ f (I(X)) and S = {A ∈ I(X) : where The pre-image of the arbitrary union and intersection of IVFI sets is just the union and intersection of the pre-images as proved below.Theorem 6.Let X and Y be nonempty sets and I(Y ) be an ideal on Y .Moreover, let f : X → Y be a mapping and { ι j : j ∈ J} be a collection of IVFI sets in I I(Y ) .Then The importance of the following concept will be seen when dealing with continuity with respect to IVFI sets.Due to Theorem 2, we have the following result which is an instance of the difference of the IVFI topology from the usual topology.Theorem 9. Let ι be an IVFI set in an IVFI space (I(X), T ).Then, int ι (cl ι c ) c and cl ι (int ι c ) c .Proof.Let ι be an IVFI set in an IVFI space (I(X), T ).Then We next make precise what we mean by continuity with respect to IVFI sets.Let f : X → Y be a one-to-one map and I(X) be an ideal on X.Then, f (I(X)) is an ideal on Y , by Theorem 3, and by Proposition 2, f −1 (f (I(X))) = I(X).Definition 8. Let f : X → Y be a one-to-one map and I(X) be an ideal on X.Let T 1 and T 2 be IVFI topologies on I(X) and f (I(X)), respectively.The map f is said to be IVFI continuous if f −1 [ ι] ∈ T 1 , for all ι ∈ T 2 .(iii) ⇐⇒ (iv) Suppose that for each IVFI point P α,B , the inverse of every neighborhood of f [P α,B ] under f is a neighborhood of P α,B .Let P α,B be an IVFI point in I I(X) and η be a neighborhood of f [P α,B ].Then by assumption, there is a neighborhood f −1 [ η] of P α,B .Take Conversely, suppose that for each IVFI point P α,B and each neighborhood η of f [P α,B ], there is a neighborhood η of P α,B such that f [ η ] = η whenever f is onto.Let P α,B be an IVFI point in I I(X) and η be a neighborhood of f [P α,B ].Then by assumption, there is a neighborhood η of P α,B such that f [ η ] = η.But note that by Theorem 5 (v), (iv) ⇐⇒ (i) Suppose that for each IVFI point P α,B and each neighborhood η of f [P α,B ], there is a neighborhood η of P α,B such that f [ η ] = η whenever f is onto.Let η ∈ T 2 and P α,B be an IVFI point in I I(X) .By assumption, η is a neighborhood of f [P α,B ].Thus, there exists a neighborhood η of P α,B such that f [ η ] = η.Taking the inverse images, Let B ∈ f (I(X)) and S = {A ∈ I(X) : f (A) = B}.If B = ∅, then S = {∅}, and so sup A∈S ι(A) = [0, 0].Also, if ∅ = A ∈ f −1 (I(Y )), then f (A) = ∅ and so τ (f (A)) = [0, 0].Hence, we have the following remark.