EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 2, 2019, 553-570 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global On Interval-Valued Fuzzy on Ideal Sets Mary Joy S. Togonon1, Randy L. Caga-anan2,∗ 1 Bukidnon State University-Baungon Satellite Campus, Bukidnon, Philippines 2 Department of Mathematics and Statistics, College of Science and Mathematics, and Premier Research Institute of Science and Mathematics, MSU-Iligan Institute of Technology, Iligan City, Philippines Abstract. 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 interval- valued 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 Key Words and Phrases: Fuzzy sets, interval-valued, ideal 1. 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, ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v12i2.3418 Email addresses: maryjoy.togonon@g.msuiit.edu.ph (MJ Togonon), randy.caga-anan@g.msuiit.edu.ph (R. Caga-anan) http://www.ejpam.com 553 c© 2019 EJPAM All rights reserved. MJ Togonon, R Caga-anan / Eur. J. Pure Appl. Math, 12 (2) (2019), 553-570 554 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 II(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 IP(X) by associating it with the fuzzy on ideal set µα defined by µα(A) = { α(x), if A = {x}, x ∈ X 0, otherwise, 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 mem- bership 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 MJ Togonon, R Caga-anan / Eur. J. Pure Appl. Math, 12 (2) (2019), 553-570 555 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. 2. 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 = [α−, α+]. Let α1 = [α−1 , α + 1 ] and α2 = [α−2 , α + 2 ] be closed subintervals of I . We use the inequality notation “≤I”, say α1 ≤I α2, to mean α−1 ≤ α − 2 and α+ 1 ≤ α + 2 . Let A be an index set and αi ∈ I , for each i ∈ A. We define the supremum of αi by sup i∈A αi = [sup i∈A α−i , sup i∈A α+ i ] and the infimum of αi by inf i∈A αi = [inf i∈A α−i , inf i∈A α+ i ]. 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 ι̂ : I(X) → I that satisfies the following: i. ι̂(∅) = [0, 0]; and ii. for nonempty sets A,B ∈ I(X) with A ⊆ B, ι̂(B) ≤I ι̂(A). We denote the set of all such ι̂ 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 π̂ : P(X)→ I by π̂(A) = { [0, 0], if A = ∅; infx∈A π(x), if A 6= ∅, A ∈ P(X). This is similar to the guaranteed possibility given in [1]. We call an IVFI set ι̂ : I(X) → I with the property that for all nonempty set A ∈ I(X), ι̂(A) = inf x∈A ι̂({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 0̃I(X) the IVFI set 0̃I(X) : I(X)→ {[0, 0]} and by 1̃I(X) the IVFI set 1̃I(X)(A) = { [0, 0], if A = ∅; [1, 1], if A 6= ∅, A ∈ I(X). MJ Togonon, R Caga-anan / Eur. J. Pure Appl. Math, 12 (2) (2019), 553-570 556 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 ι̂, τ̂ ∈ I I(X). We say (i) ι̂ is a subset of τ̂ , denoted by ι̂ 6 τ̂ , if ι̂(A) ≤I τ̂(A), for all A ∈ I(X); and (ii) ι̂ is equal to τ̂ , denoted by ι̂ = τ̂ , if ι̂(A) = τ̂(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 “6” 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 ι̂ ∈ I I(X). The complement of ι̂, denoted by ι̂c, is defined by, ι̂c(∅) = [0, 0] and for every nonempty set A ∈ I(X), ι̂c(A) = [ inf x∈A { 1− [̂ι({x})]+ } , inf x∈A { 1− [̂ι({x})]− }] . Proposition 1. Let X be a nonempty set and I(X) be an ideal on X. If ι̂ ∈ I I(X), then the complement of ι̂ is an IVFI set. Proof. Let ι̂ ∈ I I(X). Let A ∈ I(X), x ∈ A and ι̂({x}) = [[̃ι({x})]−, [̂ι({x})]+]. Since [̂ι({x})]− ≤ [̂ι({x})]+, we have 1− [̂ι({x})]+ ≤ 1− [̂ι({x})]−. Hence, inf x∈A { 1− [̂ι({x})]+ } ≤ inf x∈A { 1− [̂ι({x})]− } , and indeed we have a closed interval. We need to show that the reverse inequality of an IVFI set holds. Let ∅ 6= A,B ∈ I(X) such that A ⊆ B. Then, {1− [̂ι({x})]− : x ∈ A} ⊆ {1− [̂ι({x})]− : x ∈ B} and {1− [̂ι({x})]+ : x ∈ A} ⊆ {1− [̂ι({x})]+ : x ∈ B}. Hence, ι̂c(B) = [ inf x∈B { 1− [̂ι({x})]+ } , inf x∈B { 1− [̂ι({x})]− }] ≤I [ inf x∈A { 1− [̂ι({x})]+ } , inf x∈A { 1− [̂ι({x})]− }] = ι̂c(A). Therefore, ι̂c 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. MJ Togonon, R Caga-anan / Eur. J. Pure Appl. Math, 12 (2) (2019), 553-570 557 Definition 4. Let X be a nonempty set and I(X) be an ideal on X. Let ι̂, τ̂ ∈ I I(X). The union and intersection of ι̂ and τ̂ , denoted by ι̂∨ τ̂ and ι̂∧ τ̂ , respectively, are given by (ι̂ ∨ τ̂)(A) = [max{[̂ι(A)]−, [τ̂(A)]−},max{[̂ι(A)]+, [τ̂(A)]+}] and (ι̂ ∧ τ̂)(A) = [min{[̂ι(A)]−, [τ̂(A)]−},min{[̂ι(A)]+, [τ̂(A)]+}], for all A ∈ I(X), respectively. In general, the union and intersection of a collection of IVFI sets {ι̂j : j ∈ J}, denoted by ∨ j∈J ι̂j and ∧ j∈J ι̂j , are given by∨ j∈J ι̂j  (A) = [sup{[̂ιj(A)]− : j ∈ J}, sup{[̂ιj(A)]+ : j ∈ J}] and ∧ j∈J ι̂j  (A) = [inf{[̂ιj(A)]− : j ∈ J}, inf{[̂ιj(A)]+ : j ∈ J}], 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. Theorem 1. Let X be a nonempty set and I(X) be an ideal on X. Let ι̂, τ̂ , η̂ ∈ I I(X). Then, i. (Commutativity): ι̂ ∨ τ̂ = τ̂ ∨ ι̂ and ι̂ ∧ τ̂ = τ̂ ∧ ι̂. ii. (Associativity): (ι̂ ∨ τ̂) ∨ η̂ = ι̂ ∨ (τ̂ ∨ η̂) and (ι̂ ∧ τ̂) ∧ η̂ = ι̂ ∧ (τ̂ ∧ η̂). iii. (Transitivity): If ι̂ 6 τ̂ and τ̂ 6 η̂, then ι̂ 6 η̂. iv. (Distributivity): ι̂ ∨ (τ̂ ∧ η̂) = (ι̂ ∨ τ̂) ∧ (ι̂ ∨ η̂) and ι̂ ∧ (τ̂ ∨ η̂) = (ι̂ ∧ τ̂) ∨ (ι̂ ∧ η̂) v. (De Morgan’s Law): (ι̂ ∨ τ̂)c = ι̂c ∧ τ̂ c and (ι̂ ∧ τ̂)c = ι̂c ∨ τ̂ c Proof. Let ι̂, τ̂ , η̂ ∈ I I(X). Properties (i) and (ii) follows from the commutativity and associativity of the maximum and minimum operations. Suppose that ι̂ 6 τ̂ and τ̂ 6 η̂. Then for all A ∈ I(X), ι̂(A) ≤I τ̂(A) and τ̂(A) ≤I η̂(A), which implies that ι̂(A) ≤I η̂(A), for all A ∈ I(X). That is, ι̂ 6 η̂, easily proving (iii). To prove (iv), let A ∈ I(X) and consider that (ι̂ ∨ (τ̂ ∧ η̂))(A) = [max{[̂ι(A)]−, [(τ̂ ∧ η̂)(A)]−},max{[̂ι(A)]+, [(τ̂ ∧ η̂)(A)]+}] = [max{[̂ι(A)]−,min{[τ̂(A)]−, [η̂(A)]−}},max{[̂ι(A)]+,min{[τ̂(A)]+, [η̂(A)]+}}]. MJ Togonon, R Caga-anan / Eur. J. Pure Appl. Math, 12 (2) (2019), 553-570 558 We consider two cases. First, if τ̂(A) ≤I η̂(A), then (ι̂ ∨ (τ̂ ∧ η̂))(A) = [max{[̂ι(A)]−, [τ̂(A)]−},max{[̂ι(A)]+, [τ̂(A)]+}]. Second, if η̂(A) ≤I τ̂(A), then (ι̂ ∨ (τ̂ ∧ η̂))(A) = [max{[̂ι(A)]−, [η̂(A)]−},max{[̂ι(A)]+, [η̂(A)]+}]. Summarizing the two cases, we have (ι̂ ∨ (τ̂ ∧ η̂))(A) = [min{max{[̂ι(A)]−, [τ̂(A)]−},max{[̂ι(A)]−, [η̂(A)]−}}, min{max{[̂ι(A)]+, [τ̂(A)]+},max{[̂ι(A)]+, [η̂(A)]+}}] = ((ι̂ ∨ τ̂) ∧ (ι̂ ∨ η̂))(A). Thus, ι̂∨ (τ̂ ∧ η̂) = (ι̂∨ τ̂)∧ (ι̂∨ η̂). Similarly, we can show that ι̂∧ (τ̂ ∨ η̂) = (ι̂∧ τ̂)∨ (ι̂∧ η̂). To prove (v), let ∅ 6= A ∈ I(X) and note that (ι̂ ∨ τ̂)c(A) = [ inf x∈A { 1− [(ι̂ ∨ τ̂)({x})]+ } , inf x∈A { 1− [(ι̂ ∨ τ̂)({x})]− }] = [ inf x∈A { 1−max{[̂ι({x})]+, [τ̂({x})]+} } , inf x∈A { 1−max{[̂ι({x})]−, [τ̂({x})]−} }] . We also consider two cases. First, if τ̂({x}) ≤I ι̂({x}), then (ι̂ ∨ τ̂)c(A) = [ inf x∈A { 1− [̂ι({x})]+ } , inf x∈A { 1− [̂ι({x})]− }] . Second, if ι̂({x}) ≤I τ̂({x}), then (ι̂ ∨ τ̂)c(A) = [ inf x∈A { 1− [τ̂({x})]+ } , inf x∈A { 1− [τ̂({x})]− }] . Combining the two cases, we have (ι̂ ∨ τ̂)c(A) = [ min { inf x∈A { 1− [̂ι({x})]+ } , inf x∈A { 1− [τ̂({x})]+ }} , min { inf x∈A { 1− [̂ι({x})]− } , inf x∈A { 1− [τ̂({x})]− }}] = (ι̂c ∧ τ̂ c)(A). Hence, (ι̂∨ τ̂)c = ι̂c∧ τ̂ c. Using the same argument, we can also show that (ι̂∧ τ̂)c = ι̂c∨ τ̂ 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 ι̂, τ̂ ∈ I I(X). Then, i. ι̂ 6 (ι̂c)c; MJ Togonon, R Caga-anan / Eur. J. Pure Appl. Math, 12 (2) (2019), 553-570 559 ii. ι̂ = (ι̂c)c if and only if ι̂ is a guaranteed possibility IVFI set; and iii. if ι̂ 6 τ̂ , then τ̂ c 6 ι̂c. Proof. Let ι̂ ∈ I I(X) and ∅ 6= A ∈ I(X). Note first that for singleton sets {x} ∈ I(X), we have ι̂c({x}) = [1− [̂ι({x})]+, 1− [̂ι({x})]−]. Then, consider that (ι̂c)c(A) = [ inf x∈A { 1− [̂ιc({x})]+ } , inf x∈A { 1− [̂ιc({x})]− }] = [ inf x∈A { 1− (1− [̂ι({x})]−) } , inf x∈A { 1− (1− ι̂[({x})]+) }] = [ inf x∈A [̂ι({x})]−, inf x∈A [̂ι({x})]+ ] . (1) Since ι̂ is an IVFI set, the reverse inequality property implies that ι̂(A) = [[̂ι(A)]−, [̂ι(A)]+] ≤I [ inf x∈A [̂ι({x})]−, inf x∈A [̂ι({x})]+ ] . Thus, ι̂ 6 (ι̂c)c, proving (i). To prove (ii), recall first the definition of a guaranteed possibility IVFI set after Example 1. Now, suppose that ι̂ = (ι̂c)c. Then, ι̂(A) = (ι̂c)c(A), for all A ∈ I(X). Note from (1) that for ∅ 6= A ∈ I(X), (ι̂c)c(A) = [ inf x∈A [̂ι({x})]−, inf x∈A [̂ι({x})]+ ] . Thus, ι̂(A) = [ infx∈A [̂ι({x})]−, infx∈A [̂ι({x})]+ ] = infx∈A ι̂({x}). That is, ι̂ is a guaran- teed possibility IVFI set. Conversely, suppose that ι̂(A) = [ inf x∈A ι̂({x})−, inf x∈A ι̂({x})+ ] , for ∅ 6= A ∈ I(X). Then by (1), ι̂(A) = (ι̂c)c(A). Thus, ι̂ = (ι̂c)c. To prove (iii), suppose that ι̂ 6 τ̂ . Let ∅ 6= A ∈ I(X). Then, ι̂({x}) ≤I τ̂({x}), for all x ∈ A. Thus, for every x ∈ A, [ 1− [τ̂({x})]+, 1− [τ̂({x})]− ] ≤I [ 1− [̂ι({x})]+, 1− [̂ι({x})]− ] . Hence,[ inf x∈A { 1−[τ̂({x})]+ } , inf x∈A { 1−[τ̂({x})]− }] ≤I [ inf x∈A { 1− [̂ι({x})]+ } , inf x∈A { 1− [̂ι({x})]− }] . Thus, τ̂ c(A) ≤I ι̂c(A), for all A ∈ I(X). Therefore, τ̂ c 6 ι̂c. 3. 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) : A ∈ I(X)} and f−1(I(Y )) = {A : A ⊆ f−1(B), B ∈ I(Y )}, 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 pre- image of ideals are also ideals. The proof can be found in [6]. MJ Togonon, R Caga-anan / Eur. J. Pure Appl. Math, 12 (2) (2019), 553-570 560 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. i. If ι̂ ∈ I I(X), then the image of ι̂ under f , denoted by f [̂ι], is the mapping f [̂ι] : f(I(X))→ I given by (f [̂ι])(B) = [ sup A∈S [̂ι(A)]−, sup A∈S [̂ι(A)]+ ] , where S = {A ∈ I(X) : f(A) = B}. ii. If τ̂ ∈ I I(Y ), then the pre-image of τ̂ under f , denoted by f−1[τ̂ ], is the mapping f−1[τ̂ ] : f−1(I(Y ))→ I given by (f−1[τ̂ ])(A) = (τ̂ ◦ f)(A), where (τ̂ ◦ f)(A) is the composition τ̂(f(A)). Let B ∈ f(I(X)) and S = {A ∈ I(X) : f(A) = B}. If B = ∅, then S = {∅}, and so sup A∈S ι̂(A) = [0, 0]. Also, if ∅ = A ∈ f−1(I(Y )), then f(A) = ∅ and so τ̂(f(A)) = [0, 0]. Hence, we have the following remark. Remark 5. Let X and Y be nonempty sets and f : X → Y be a mapping. Let ι̂ and τ̂ be IVFI sets in I I(X) and I I(Y ), respectively. Then, we have f [̂ι](∅) = [0, 0] and f−1[τ̂ ](∅) = [0, 0]. Theorem 4. Let X and Y be nonempty sets and f : X → Y be a mapping. Let ι̂ and τ̂ be IVFI sets defined on the ideals I(X) and I(Y ), respectively. Then, f [̂ι] and f−1[τ̂ ] 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 ∅ 6= B1, B2 ∈ f(I(X)) such that B1 ⊆ B2. Let S1 = {A ∈ I(X) : f(A) = B1} and S2 = {A ∈ I(X) : f(A) = B2}. Since I(X) is an ideal, for every A ∈ I(X) such that f(A) = B2, there exists A1 ∈ I(X) such that A1 ⊆ A and f(A1) = B1. Since ι̂ is an IVFI set and A1 ⊆ A, we have ι̂(A) ≤I ι̂(A1). Hence, supA∈S2 ι̂(A) ≤I supA1∈S1 ι̂(A1). Thus, f [̂ι](B2) ≤I f [̂ι](B1). 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−1[τ̂ ] is an IVFI set defined on the ideal f−1(I(Y )), let A,A1 ∈ f−1(I(Y )) such that A1 ⊆ A. Then, f(A1) ⊆ f(A) and f(A1), f(A) ∈ I(Y ). Since τ̂ is an IVFI set on I(Y ), τ̂(f(A)) ≤I τ̂(f(A1)). Thus, MJ Togonon, R Caga-anan / Eur. J. Pure Appl. Math, 12 (2) (2019), 553-570 561 f−1[τ̂ ](A) ≤I f−1[τ̂ ](A1). Therefore, with Remark 5 and Theorem 3, f−1[τ̂ ] 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 ι̂ ∈ I I(X) and η̂ ∈ I I(Z), then (g ◦ f)[̂ι] and (g ◦ f)−1(η̂) are IVFI sets defined on (g ◦ f)(I(X)) and (g ◦ f)−1(I(Z)), respectively. 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. f(f−1(I(Y ))) = I(Y ), if f is onto; and ii. f−1(f(I(X))) = I(X), if f is one-to-one. Proof. Suppose that f is onto. Let B ∈ f(f−1(I(Y ))). Then there exists A ∈ f−1(I(Y )) such that f(A) = B. Since A ∈ f−1(I(Y )), A ⊆ f−1(B1) for some B1 ∈ I(Y ). Note that f(A) ⊆ f(f−1(B1)) = B1, since f is onto. Thus, B ⊆ B1. By the definition of an ideal, B ∈ I(Y ). Hence, f(f−1(I(Y ))) ⊆ I(Y ). Conversely, let B ∈ I(Y ) and C = f−1(B). Then C ∈ f−1(I(Y )). We thus have f(C) ∈ f(f−1(I(Y ))). Since f is onto, B = f(f−1(B)) = f(C) ∈ f(f−1(I(Y ))). Thus, I(Y ) ⊆ f(f−1(I(Y ))). Therefore, f(f−1(I(Y ))) = I(Y ). Suppose that f is one-to-one. Let A ∈ f−1(f(I(X))). Then there exists B ∈ f(I(X)) such that f−1(B) = A. Since B ∈ f(I(X)), B = f(A1) for some A1 ∈ I(X). Since f is one-to-one, we have A = f−1(B) = f−1(f(A1)) = A1 . Thus, A ∈ I(X). Hence, f−1(f(I(X))) ⊆ I(X). Conversely, let A ∈ I(X) and D = f(A). Then, D ∈ f(I(X)). We thus have f−1(D) ∈ f−1(f(I(X))). Since f is one-to-one, A = f−1(f(A)) = f−1(D) ∈ f−1(f(I(X))). Thus, I(X) ⊆ f−1(f(I(X))). Therefore, f−1(f(I(X))) = I(X). 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 ι̂, τ̂ ∈ I I(X) and ω̂, η̂ ∈ I I(Y ), then i. f−1 [η̂c] = (f−1 [η̂])c; ii. (f [̂ι])c 6 f [̂ιc]; MJ Togonon, R Caga-anan / Eur. J. Pure Appl. Math, 12 (2) (2019), 553-570 562 iii. if ω̂ 6 η̂, then f−1[ω̂] 6 f−1[η̂]; iv. if ι̂ 6 τ̂ , then f [̂ι] 6 f [τ̂ ]; v. if f is onto, then f [f−1[η̂]] = η̂; and vi. if f is one-to-one, then ι̂ 6 f−1[f [̂ι]]. Proof. i. Let ∅ 6= A ∈ f−1(I(Y )). Then (f−1[η̂])c(A) = [ inf x∈A { 1− [(f−1[η̂])({x})]+ } , inf x∈A { 1− [(f−1[η̂])({x})]− }] = [ inf x∈A { 1− [(η̂ ◦ f)({x})]+ } , inf x∈A { 1− [(η̂ ◦ f)({x})]− }] = [ inf x∈A { 1− [η̂(f({x}))]+ } , inf x∈A { 1− [η̂(f({x}))]− }] = [ inf f(x)∈f(A) { 1− [η̂(f({x}))]+ } , inf f(x)∈f(A) { 1− [η̂(f({x}))]− }] = [[η̂c(f(A))]−, [η̂c(f(A))]+] = [[(η̂c ◦ f)(A)]−, [(η̂c ◦ f)(A)]+] = [[(f−1[η̂c])(A)]−, [(f−1[η̂c])(A)]+] = f−1[η̂c](A). Hence, (f−1[η̂])c = f−1[η̂c]. ii. Let ∅ 6= B ∈ f(I(X)) and S = {A ∈ I(X) : f(A) = B}. Then (f [̂ι])c(B) = [ inf y∈B { 1− [f [̂ι]({y})]− } , inf y∈B { 1− [f [̂ι]({y})]+ }] = [ inf y∈B { 1− sup A∈S′ [̂ι(A)]− } , inf y∈B { 1− sup A∈S′ [̂ι(A)]+ }] ; where S′ = {A ∈ I(X) : f(A) = {y}}. Since 1− sup A∈S′ ι̂(A) = inf A∈S′ {1− ι̂(A)}, we have (f [̂ι])c(B) = [ inf y∈B { inf A∈S′ { 1− [̂ι(A)]− }} , inf y∈B { inf A∈S′ { 1− [̂ι(A)]+ }}] . Let S′y = {{x} ∈ I(X) : f({x}) = {y}}. Observe that since ι̂ is an IVFI set, if {x} ⊆ A, then ι̂(A) ≤I ι̂({x}) and 1− ι̂({x}) ≤I 1− ι̂(A). Noting that S′y ⊆ S′, we thus have[ inf A∈S′ { 1− [̂ι(A)]− } , inf A∈S′ { 1− [̂ι(A)]+ }] = [ inf A∈S′y { 1− [̂ι(A)]− } , inf A∈S′y { 1− [̂ι(A)]+ }] . Hence, (f [̂ι])c(B) = [ inf y∈B { inf {x}∈S′y { 1− [̂ι({x})]− }} , inf y∈B { inf {x}∈S′y { 1− [̂ι({x})]+ }}] . MJ Togonon, R Caga-anan / Eur. J. Pure Appl. Math, 12 (2) (2019), 553-570 563 Consider that[ inf y∈B { inf {x}∈S′y { 1− [̂ι({x})]− }} , inf y∈B { inf {x}∈S′y { 1− [̂ι({x})]+ }}] = [ inf x∈AB { 1− [̂ι({x})]− } , inf x∈AB { 1− [̂ι({x})]+ }] = [[̂ιc(AB)]−, [̂ιc(AB)]+], where AB = {x : {x} ∈ S′y , y ∈ B}. Thus, (f [̂ι])c(B) = [[̂ιc(AB)]−, [̂ιc(AB)]+]. Note that AB ∈ S. Then, (f [̂ι])c(B) = [[̂ιc(AB)]−, [̂ιc(AB)]+] ≤I [ sup A∈S [̂ιc(A)]−, sup A∈S [̂ιc(A)]+ ] = [[f [̂ιc](B)]−, [f [̂ιc](B)]+] = f [̂ιc](B). Hence, (f [̂ι])c 6 f [̂ιc]. iii. Let A ∈ f−1(I(Y )). Then, A ⊆ f−1(B), for some B ∈ I(Y ). Note that f(A) ⊆ f(f−1(B)) ⊆ B ∈ I(Y ). Thus, f(A) ∈ I(Y ). If ω̂ 6 η̂, then ω̂(f(A)) ≤I η̂(f(A)). Consider that (f−1[ω̂])(A) = (ω̂ ◦ f)(A) = ω̂(f(A)) ≤I η̂(f(A)) = (η̂ ◦ f)(A) = (f−1[η̂])(A). Therefore, f−1[ω̂] 6 f−1[η̂]. iv. Let B ∈ f(I(X)) and S = {A ∈ I(X) : f(A) = B}. If ι̂ 6 τ̂ , then ι̂(A) ≤I τ̂(A), for all A ∈ I(X). Hence, (f [̂ι])(B) = [ sup A∈S [̂ι(A)]−, sup A∈S [̂ι(A)]+ ] ≤I [ sup A∈S [τ̂(A)]−, sup A∈S [τ̂(A)]+ ] = (f [τ̂ ])(B). Therefore, f [̂ι] 6 f [τ̂ ]. v. Suppose that f is onto. Let B ∈ f(f−1(I(Y ))) = I(Y ) and S = {A ∈ f−1(I(Y )) : f(A) = B}. Then f [f−1[η̂]](B) = [ sup A∈S [(f−1[η̂])(A)]−, sup A∈S [(f−1[η̂])(A)]+ ] = [ sup A∈S [(η̂ ◦ f)(A)]−, sup A∈S [(η̂ ◦ f)(A)]+ ] = [ sup A∈S [η̂(f(A))]−, sup A∈S [η̂(f(A))]+ ] MJ Togonon, R Caga-anan / Eur. J. Pure Appl. Math, 12 (2) (2019), 553-570 564 = [[η̂(B)]−, [η̂(B)]+] = η̂(B). Thus, f [f−1[η̂]] = η̂. vi. Suppose that f is one-to-one. Let A ∈ f−1(f(I(X))) = I(X). Then, A ⊆ f−1(B), for some B ∈ f(I(X)). Hence, f(A) ⊆ B. Since f(I(X)) is an ideal, f(A) ∈ f(I(X)). Then, f−1[f [̂ι]](A) = (f [̂ι] ◦ f)(A) = f [̂ι](f(A)) = [ sup C∈S [̂ι(C)]−, sup C∈S [̂ι(C)]+ ] , where S = {C ∈ I(X) : f(C) = f(A)}. Since A ∈ S, we have ι̂(A) ≤I [ sup C∈S [̂ι(C)]−, sup C∈S [̂ι(C)]+ ] . Thus, ι̂ 6 f−1[f [̂ι]]. 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 {ι̂j : j ∈ J} be a collection of IVFI sets in I I(Y ). Then i. f−1 ∨ j∈J ι̂j  = ∨ j∈J f−1 [̂ιj ]; and ii. f−1 ∧ j∈J ι̂j  = ∧ j∈J f−1 [̂ιj ]. Proof. Let A ∈ f−1(I(Y )). Consider thatf−1 ∨ j∈J ι̂j  (A) = ∨ j∈J ι̂j ◦ f  (A) = ∨ j∈J ι̂j  (f(A)) = [ sup j∈J {[̂ιj(f(A))]−}, sup j∈J {[̂ιj(f(A))]+} ] = [ sup j∈J {[(ι̂j ◦ f)(A)]−}, sup j∈J {[(ι̂j ◦ f)(A)]+} ] MJ Togonon, R Caga-anan / Eur. J. Pure Appl. Math, 12 (2) (2019), 553-570 565 = ∨ j∈J (ι̂j ◦ f)(A) = ∨ j∈J f−1 [̂ιj ]  (A). Thus, f−1 ∨ j∈J ι̂j  = ∨ j∈J f−1 [̂ιj ], proving (i). Result (ii) can be proved similarly. 4. Topology and continuity With the operations on IVFI sets, we can have an analogue of the classical topology. Definition 6. Let X be a nonempty set and I(X) be an ideal on X. An IVFI topology is a family T ′ of IVFI sets such that: (i) 0̃I(X), 1̃I(X) ∈ T ′; (ii) if ι̂, τ̂ ∈ T ′, then ι̂ ∧ τ̂ ∈ T ′; and (iii) if {ι̂j : j ∈ J} ⊆ T ′, then ∨ j∈J ι̂j ∈ T ′. We call the ordered pair (I(X),T ′) an IVFI space and an element of T ′ an IVFI open set. An IVFI set ι̂ will be called IVFI closed if its complement is IVFI open. It can be easily seen that if {T ′α : α ∈ A } is a family of IVFI topologies on I(X), then⋂ α∈A T ′α is also an IVFI topology. However, ⋃ α∈A T ′α need not be. Let (I(X),T ′) be an IVFI space. We call the subcollection B of T ′ an IVFI base for T ′ if every member of T ′ can be expressed as a union of members of B. Let [0, 0] 6= α ∈ I and ∅ 6= B ∈ I(X). We call the IVFI set given by Pα,B(A) = { α , if A ⊆ B,A 6= ∅; [0, 0] , otherwise. an IVFI point. One can calculate that the explicit form of the complement of Pα,B is given by P cα,B(A) =  [0, 0] , if A = ∅; [1− α+, 1− α−] , if A ∩B 6= ∅; [1, 1] , if A 6= ∅ and A ∩B = ∅. We say that Pα,B is contained in an IVFI set ι̂, denoted by Pα,B ∈ ι̂, if and only if α ≤I ι̂(B). With this, we can characterize an IVFI base and IVFI open sets. Remark 6. Every IVFI set ι̂ can be expressed as the union of all IVFI points which is contained in ι̂. That is, if ι̂(B) is not zero for B ∈ I(X), then ι̂(B) = sup{α : Pα,B is an IVFI point and [0, 0]