The Properties of Intuitionistic Anti Fuzzy Module t-norm and t-conorm CAUCHY –Jurnal Matematika Murni dan Aplikasi Volume 7(2) (2022), Pages 207-219 p-ISSN: 2086-0382; e-ISSN: 2477-3344 Submitted: September 15, 2021 Reviewed: December 10, 2021 Accepted: February 20, 2022 DOI: http://dx.doi.org/10.18860/ca.v7i1.13351 The Properties of Intuitionistic Anti Fuzzy Module t-norm and t- conorm Ongky Denny Wijaya, Abdul Rouf Alghofari, Noor Hidayat, Mohamad Muslikh* Mathematics Department Brawijaya University, Malang, Indonesia *Corresponding Author Email: mslk@ub.ac.id* ongkydenny@gmail.com, abdul_rouf@ub.ac.id, noorh@ub.ac.id ABSTRACT Zadeh have introduced fuzzy set in 1965 and Atanassov have introduced intuitionistic fuzzy set in 1986 in theirs paper. Now, many of researcher connecting intuitionistic fuzzy set with algebra theory. We interested to combine some concepts in references over intuitionistic fuzzy set, module of a ring, t-norm, t-conorm, and intuitionistic anti fuzzy. In this paper, we discuss about intuitionistic anti fuzzy module t-norm and t-conorm (IAFMTC) and their properties with respect to module homomorphism, maps, pre-image, and anti-image from intuitionistic fuzzy sets. We have investigated and prove all general properties of IAFMTC and properties related to module homomorphism, maps, pre-image, and anti-image. Keywords: Intuitionistic Fuzzy Set; t-norm; t-conorm; Module INTRODUCTION In 1965, Zadeh [1] introduced new concept about fuzzy set. This concept highlighting the membership status of an indeterminate or fuzzy set. In this concept, membership status is defined as a function whose value is in the interval [0, 1]. In 1986, Atanassov [2] introduced the notion of intuitionistic fuzzy sets as generalization of fuzzy sets, i.e. highlight membership function and non membership function which the value sum of both functions lies in [0,1]. Today, much of the research in the field of algebra was utilised with intuitionistic fuzzy sets on their works. As on the article which written by Isaac and John [3] they introduce intuitionistic fuzzy module and their properties. Next, Rahman and Saikia [4] was introduced a concept of t-norms with respect to the Intuitionistic Fuzzy Submodule, then he describes some properties. In other side, Rasuli [5] was introduced the concepts and properties of intuitionistic fuzzy vector space with respect to t-norm and t-conorm. In contrast to the results above, Sharma in his article [6] introduces a submodule of the intuitionistic anti-fuzzy module. Sharma also discusses the concept of intuitionistic anti- fuzzy modules and the properties it gives rise to. In [7] Sharma was discussed about several properties in maps, pre-image, and anti-image in intuitionistic fuzzy module. A function called t-norm is studied on statistical metric [8] and both of t-norm and its dual, i.e. t-conorm is studied on probabilistic metric spaces [9]. Before many research in http://dx.doi.org/10.18860/ca.v7i1.13351 mailto:mslk@ub.ac.id mailto:ongkydenny@gmail.com mailto:abdul_rouf@ mailto:noorh@ The Properties of Intuitionistic Anti Fuzzy Module t-norm and t-conorm Ongky Denny Wijaya 208 intuitionistic fuzzy sets related with algebra, there are also many research in fuzzy sets related with algebra. Some of them are fuzzy modules over a t-norm by Rasuli in [10] and anti-fuzzy submodule of a module by Sharma in [11]. Some properties about homomorphism in group algebraic structure related by intuitionistic fuzzy was discussed by Sharma in [12]. Motivated by the results of [3-12] in this article, we introduced some relatively new concepts on intuitionistic anti fuzzy module with respect to t-norm and t-conorm (IAFMTC) as opposite of intuitionistic fuzzy submodule related by t-norm and t-conorm. This concept combines some concepts from the results mentioned above as the gap of previous research. We investigate general properties on IAFMTC and properties with respect to module homomorphism, maps, pre-image, and anti-image from intuitionistic fuzzy sets. We give also some examples to illustrate the main results in this article. METHODS We make literature review from [3-12]. Then we construct a new structure named intuitionistic anti fuzzy module t-norm and t-conorm by modifying definition intuitionistic anti fuzzy module (IAFM) from [6] or intuitionistic fuzzy module w.r.t t- norm and t-conorm (IFMTC) from [4]. We have modified definition of IAFM by change minimum value with t-norm and maximum value using t-conorm. Also, we have modified definition of IFMTC by change “less than or equal” sign with “greater than or equal” sign. RESULTS AND DISCUSSION In this paper let 𝑅 denotes commutative ring with unity 1. Definition of ring see [13] or [14]. Now, this is definition of module over 𝑅. Definition 1. [14] Let 𝑅 be a ring with identity 1. An abelian group 𝑀 said to be a module over 𝑅 (denoted by 𝑅-module) if the map ⋅: 𝑅 × 𝑀 → 𝑀 (𝑟, 𝑚) ↦⋅ (𝑟, 𝑚) = 𝑟𝑚 satisfying the four conditions: (1) 𝑟(𝑚1 + 𝑚2) = 𝑟𝑚1 + 𝑟𝑚2, (2) (𝑟1 + 𝑟2)𝑚 = 𝑟1𝑚 + 𝑟2𝑚, (3) (𝑟1𝑟2)𝑚 = 𝑟1(𝑟2𝑚), (4) 1𝑚 = 𝑚, for all 𝑟, 𝑟1, 𝑟2 ∈ 𝑅 and 𝑚, 𝑚1, 𝑚2 ∈ 𝑀. For the future, module over a ring 𝑅 will be denoted by 𝑅-module. For a submodule of a module have simple criteria as defined as below. Definition 2. [14] Let 𝑀 be a 𝑅-module and 𝑁 be a non empty subset of 𝑀. 𝑁 is said to be submodule of 𝑀 if two conditions below is satisfied: (1) 𝑁 is a subgroup of 𝑀, i.e. for all 𝑎, 𝑏 ∈ 𝑁 then 𝑎 − 𝑏 ∈ 𝑁. (2) For all 𝑛 ∈ 𝑁 and 𝑟 ∈ 𝑅, 𝑟𝑛 ∈ 𝑁. The Properties of Intuitionistic Anti Fuzzy Module t-norm and t-conorm Ongky Denny Wijaya 209 The homomorphism maps on the module not much different on the other algebraic structures. For instance, homomorphism in vector spaces (we called it by linear maps, see [15]) is relatively similar with homomorphism maps in module. Now, definition of homomorphism maps on the module given in Definition 3. Definition 3. [14] Let 𝑀 and 𝑁 be a 𝑅-module and 𝑓: 𝑀 → 𝑁 is a map. 𝑓 is said to be 𝑅-module homomorphism, if for all 𝑎, 𝑏 ∈ 𝑀 and 𝑟 ∈ 𝑅 satisfy two conditions as follows. (1) 𝑓(𝑎 + 𝑏) = 𝑓(𝑎) + 𝑓(𝑏), (2) 𝑓(𝑟𝑎) = 𝑟𝑓(𝑎). Homomorphism 𝑓 is said an epimorphism if 𝑓 surjective, monomorphism if 𝑓 injective, isomorphism if 𝑓 bijective. If domain and codomain of a homomorphism 𝑓 are equal, then we said 𝑓 endomorphism. We called 𝑓 automorphisma, if 𝑓 bijective and endomorphism. Zadeh have introduced fuzzy sets as follows. Definition 4. [1] Let 𝑋 be non empty set. A fuzzy set 𝐴 in 𝑋 is characterized by a membership function 𝜇𝐴 which associates each point in 𝑋 with a real number in the interval [0,1]. In other words, we say the fuzzy sets is a set 𝐴 = {(𝑥, 𝜇𝐴(𝑥))|𝑥 ∈ 𝑋} which 𝜇𝐴: 𝑋 → [0,1]. The value of 𝜇𝐴 at 𝑥, i.e. 𝜇𝐴(𝑥) representing the degree of membership of 𝑥 in 𝐴. Atanassov [2] on 1986 have introduced the concept of intuitionistic fuzzy set as follows. This concept is considered as generalization of fuzzy sets. Definition 5. [2] Let 𝑋 be a non empty set. An intuitionistic fuzzy set (IFS) 𝐴 in 𝑋 is defined as an set of the form 𝐴 = {(𝑥, 𝜇𝐴(𝑥), 𝜈𝐴(𝑥))|𝑥 ∈ 𝑋} which 𝜇𝐴(𝑥) and 𝜈𝐴(𝑥) are the functions defined by 𝜇𝐴: 𝑋 → [0,1] and 𝜈𝐴: 𝑋 → [0,1], which 0 ≤ 𝜇𝐴(𝑥) + 𝜈𝐴(𝑥) ≤ 1 for every 𝑥 ∈ 𝑋. The functions 𝜇𝐴(𝑥) and 𝜈𝐴(𝑥) define the degree of membership and degree of non- membership respectively, for every 𝑥 ∈ 𝑋. The definition of triangular norm was introduced on [8] in 1942. In the beginning, triangular norm used in the study of probabilistic metric spaces. In [9], triangular norm and triangular conorm was studied more at probabilistic metric spaces. In the following, triangular norm and triangular conorm are defined as follows. Definition 6. [4, 5] Let 𝑇 be a function which defined as 𝑇: [0,1] × [0,1] → [0,1]. 𝑇 is said to be a triangular norm (denoted by t-norm) if for all 𝑥, 𝑦, 𝑧 ∈ [0,1], four axioms are hold: (1) Neutral element, i.e. 𝑇(𝑥, 1) = 𝑥, (2) Monotonicity, i.e. if 𝑦 ≤ 𝑧 then 𝑇(𝑥, 𝑦) ≤ 𝑇(𝑥, 𝑧), (3) Commutativity, i.e. 𝑇(𝑥, 𝑦) = 𝑇(𝑦, 𝑥), (4) Associativity, i.e. 𝑇(𝑥, 𝑇(𝑦, 𝑧)) = 𝑇(𝑇(𝑥, 𝑦), 𝑧). The Properties of Intuitionistic Anti Fuzzy Module t-norm and t-conorm Ongky Denny Wijaya 210 Definition 7. [4, 5] Let 𝐶 be a function which defined as 𝐶: [0,1] × [0,1] → [0,1]. 𝑇 is said to be a triangular conorm (denoted by t-conorm) if for all 𝑥, 𝑦, 𝑧 ∈ [0,1], four axioms are hold: (1) Neutral element, i.e. 𝐶(𝑥, 0) = 𝑥, (2) Monotonicity, i.e. if 𝑦 ≤ 𝑧 then 𝐶(𝑥, 𝑦) ≤ 𝐶(𝑥, 𝑧), (3) Commutativity, i.e. 𝐶(𝑥, 𝑦) = 𝐶(𝑦, 𝑥), (4) Associativity, i.e. 𝐶(𝑥, 𝐶(𝑦, 𝑧)) = 𝐶(𝐶(𝑥, 𝑦), 𝑧). Example 1. [5] The examples of t-norm and t-conorm is given as follows. Table 1. The examples of t-norm and t-conorm Name t-norm t-conorm Standard intersection/ standard union 𝑇𝑚 (𝑥, 𝑦) = min{𝑥, 𝑦} 𝐶𝑚 (𝑥, 𝑦) = max{𝑥, 𝑦} Bounded sum 𝑇𝑏 (𝑥, 𝑦) = max{0, 𝑥 + 𝑦 − 1} 𝐶𝑏 (𝑥, 𝑦) = min{1, 𝑥 + 𝑦} Algebraic product/ Algebraic sum 𝑇𝑝(𝑥, 𝑦) = 𝑥𝑦 𝐶𝑝(𝑥, 𝑦) = 𝑥 + 𝑦 − 𝑥𝑦 Drastic 𝑇𝐷 (𝑥, 𝑦) = { 𝑦 if 𝑥 = 1 𝑥 if 𝑦 = 1 0 otherwise 𝐶𝐷(𝑥, 𝑦) = { 𝑦 if 𝑥 = 0 𝑥 if 𝑦 = 0 1 otherwise Nilpotent minimum/ Nilpotent maximum 𝑇𝑛𝑀 (𝑥, 𝑦) = { min{𝑥, 𝑦} if 𝑥 + 𝑦 > 1 0 otherwise 𝐶𝑛𝑀 (𝑥, 𝑦) = { max{𝑥, 𝑦} if 𝑥 + 𝑦 < 1 1 otherwise Hamacher product/ Einstein sum 𝑇𝐻0 (𝑥, 𝑦) = { 0 if 𝑥 = 𝑦 = 0 𝑥𝑦 𝑥 + 𝑦 − 𝑥𝑦 otherwise 𝐶𝐻2 (𝑥, 𝑦) = 𝑥 + 𝑦 1 + 𝑥𝑦 Definition 8. [5] Let 𝑇 be a t-norm and 𝐶 be a t-conorm. 𝑇 is said to be idempotent t-norm and 𝐶 is said to be idempotent t-conorm if for all 𝑥 ∈ [0,1] satisfy 𝑇(𝑥, 𝑥) = 𝑥 and 𝐶(𝑥, 𝑥) = 𝑥 respectively. In [6], there are several corollaries of t-norm and t-conorm as follows. Corollary 1. [5] Let 𝑇 be a t-norm, then for all 𝑥 ∈ [0,1], (1) 𝑇(𝑥, 0) = 0, (2) 𝑇(0,0) = 0. The Properties of Intuitionistic Anti Fuzzy Module t-norm and t-conorm Ongky Denny Wijaya 211 Corollary 2. [5] Let 𝐶 be a t-conorm, then for all 𝑥 ∈ [0,1], (1) 𝐶(𝑥, 1) = 1, (2) 𝐶(0,0) = 0. Definition 9. [7] Let 𝑋 and 𝑌 are non empty sets, 𝑓: 𝑋 → 𝑌 a maps, 𝐴 = (𝜇𝐴, 𝜈𝐴) and 𝐵 = (𝜇𝐵 , 𝜈𝐵 ) are intuitionistic fuzzy sets on 𝑋 and 𝑌 respectively. (1) For all 𝑦 ∈ 𝑌, intuitionistic fuzzy sets 𝑓(𝐴) = {(𝑦, 𝜇𝑓(𝐴)(𝑦), 𝜇𝜈(𝐴)(𝑦)) |𝑦 ∈ 𝑌} which 𝜇𝑓(𝐴)(𝑦) = { max 𝑥∈𝑓−1(𝑦) 𝜇𝐴(𝑥) 𝑓 −1(𝑦) ≠ 0 0 𝑓 −1(𝑦) = 0 and 𝜈𝑓(𝐴)(𝑦) = { min 𝑥∈𝑓−1(𝑦) 𝜈𝐴(𝑥) 𝑓 −1(𝑦) ≠ 0 1 𝑓 −1(𝑦) = 0 , is called by image of 𝐴 by 𝑓. (2) For all 𝑥 ∈ 𝑋, intuitionistic fuzzy sets 𝑓 −1(𝐵)(𝑥) = (𝜇𝑓−1(𝐵)(𝑥), 𝜈𝑓−1(𝐵)(𝑥)) which 𝜇𝑓−1(𝐵)(𝑥) = 𝜇𝐵 (𝑓(𝑥)) and 𝜈𝑓−1(𝐵)(𝑥) = 𝜈𝐵 (𝑓(𝑥)), is called by pre-image 𝐴 by 𝑓. (3) For all 𝑦 ∈ 𝑌, intuitionistic fuzzy sets 𝑓(𝐴) = {(𝑦, 𝜇�̂�(𝐴)(𝑦), 𝜈�̂�(𝐴)(𝑦)) |𝑦 ∈ 𝑌} which 𝜇�̂�(𝐴)(𝑦) = { min 𝑥∈𝑓−1(𝑦) 𝜇𝐴(𝑥) 𝑓 −1(𝑦) ≠ 0 1 𝑓 −1(𝑦) = 0 and 𝜈�̂�(𝐴)(𝑦) = { max 𝑥∈𝑓−1(𝑦) 𝜈𝐴(𝑥) 𝑓 −1(𝑦) ≠ 0 0 𝑓 −1(𝑦) = 0 , is called by anti-image of 𝐴 by 𝑓. Intuitionistic fuzzy module has discussed before in [3] by Isaac and John on 2011. Let 𝑀 be a 𝑅-module and 𝐴 = (𝜇𝐴, 𝜈𝐴) is intuitionistic fuzzy set on 𝑀, then 𝐴 is called by intuitionistic fuzzy module if satisfy six axioms, i.e. (1) 𝜇𝐴(0) = 1, (2) 𝜇𝐴(𝑥 + 𝑦) ≥ min{𝜇𝐴(𝑥), 𝜇𝐴(𝑦)}, (3) 𝜇𝐴(𝑎𝑥) ≥ 𝜇𝐴(𝑥), (4) 𝜈𝐴(0) = 0, (5) 𝜈𝐴(𝑥 + 𝑦) ≤ max{𝜈𝐴(𝑥), 𝜈𝐴(𝑦)}, (6) 𝜈𝐴(𝑎𝑥) ≤ 𝜈𝐴(𝑥), for all 𝑥 ∈ 𝑀 and 𝑎 ∈ 𝑅. Intuitionistic anti fuzzy module (abbreviated by IAFM) was discussed by Sharma in [6]. Intuitionistic anti fuzzy module obtained by modifying 𝜇𝐴(0) = 1 by 𝜇𝐴(0) = 0, min by max on second and fifth axiom, sign ≥ by ≤ and vice versa. In this section, we introduce intuitionistic anti fuzzy module t-norm and t- conorm (abbreviated by IAFMTC), the example, some corollaries, and the properties related by module homomorphism, image, pre-image, and anti-image. The Properties of Intuitionistic Anti Fuzzy Module t-norm and t-conorm Ongky Denny Wijaya 212 Definition 10. Let 𝑀 be 𝑅-module and 𝐴 = (𝜇𝐴, 𝜈𝐴) be intuitionistic fuzzy set on 𝑀. 𝐴 is called by intuitionistic anti fuzzy module t-norm and t-conorm (denoted by 𝐴 ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀)) if six axioms below are holds: (1) 𝜇𝐴(0) = 0, (2) 𝜇𝐴(𝑥 + 𝑦) ≤ 𝐶(𝜇𝐴(𝑥), 𝜇𝐴(𝑦)), (3) 𝜇𝐴(𝑎𝑥) ≤ 𝜇𝐴(𝑥), (4) 𝜈𝐴(0) = 1, (5) 𝜈𝐴(𝑥 + 𝑦) ≥ 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(𝑦)), (6) 𝜈𝐴(𝑎𝑥) ≥ 𝜈𝐴(𝑥), for all 𝑥, 𝑦 ∈ 𝑀 and 𝑎 ∈ 𝑅. In following, we give example of intuitionistic anti fuzzy module t-norm and t- conorm. Example 2. Let 𝑌 is a set 𝑌 = {1,2}. The power set of 𝑌 is 𝑃(𝑌) = {∅, {1}, {2}, {1,2}}. It is easy to check 𝑃(𝑌) form a ring under the operations ⊕ (symmetric difference) and ∩ (intersection). Now, also easy to check that 𝑃(𝑌) is a 𝑃(𝑌)-module. If 𝐴 = (𝜇𝐴, 𝜈𝐴) is intuitionistic fuzzy set on 𝑃(𝑌) which 𝜇𝐴: 𝑃(𝑌) → [0,1] 𝐵 ↦ 𝜇𝐴(𝐵) = { 0 𝐵 = ∅ 0.4 𝐵 = {1} or {2} 0.5 𝐵 = {1,2} and 𝜈𝐴: 𝑃(𝑌) → [0,1] 𝐵 ↦ 𝜈𝐴(𝐵) = { 1 𝐵 = ∅ 0.3 𝐵 = {1} or {2} 0.25 𝐵 = {1,2} then 𝐴 is intuitionistic anti fuzzy module: (a) bounded sum t-norm and bounded sum t-conorm. (b) algebraic product t-norm and algebraic sum t-conorm. Proof. To prove it, we must check all conditions on Definition 10. ◼ Corollary 3. If 𝑀 is 𝑅-module and 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀) then 𝜇𝐴(−𝑥) ≤ 𝜇𝐴(𝑥) and 𝜈𝐴(−𝑥) ≥ 𝜈𝐴(𝑥), for all 𝑥 ∈ 𝑀. Proof. Since 𝑀 is 𝑅-module then for all 𝑥 ∈ 𝑀 is satisfy −𝑥 ∈ 𝑀. Based on axiom on Definition 10, we have 𝜇𝐴(𝑎(−𝑥)) = 𝜇𝐴((−𝑎)𝑥) ≤ 𝜇𝐴(𝑥) and 𝜈𝐴(𝑎(−𝑥)) = 𝜈𝐴((−𝑎)𝑥) ≥ 𝜈𝐴(𝑥). Now, choose 𝑎 as multiplication identity on 𝑅, i.e. 𝑎 = 1. We have 𝜇𝐴(−𝑥) ≤ 𝜇𝐴(𝑥) and 𝜈𝐴(−𝑥) ≥ 𝜈𝐴(𝑥). ◼ Corollary 4. If 𝑀 is 𝑅-module and 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀) then 𝐴(𝑥) = 𝐴(−𝑥), for all 𝑥 ∈ 𝑀. The Properties of Intuitionistic Anti Fuzzy Module t-norm and t-conorm Ongky Denny Wijaya 213 Proof. We prove based on axiom on Definition 10. Consider that 𝜇𝐴(−𝑥) ≤ 𝜇𝐴(𝑥) = 𝜇𝐴(−(−𝑥)) ≤ 𝜇𝐴(−𝑥) and 𝜈𝐴(−𝑥) ≥ 𝜈𝐴(𝑥) = 𝜈𝐴(−(−𝑥)) ≥ 𝜈𝐴(−𝑥). This imply 𝜇𝐴(−𝑥) = 𝜇𝐴(𝑥) dan 𝜈𝐴(−𝑥) = 𝜈𝐴(𝑥). Now, we have 𝐴(−𝑥) = (𝜇𝐴(−𝑥), 𝜈𝐴(−𝑥)) = (𝜇𝐴(𝑥), 𝜈𝐴(𝑥)) = 𝐴(𝑥). ◼ Next, we give lemma about idempotent t-norm and t-conorm. Lemma 1. 𝑇 is a idempotent t-norm if and only if 𝑇(𝑥, 𝑦) = min{𝑥, 𝑦} for all 𝑥, 𝑦 ∈ [0,1]. Proof. (⇒) Take any 𝑥, 𝑦 ∈ [0,1]. If 𝑥 ≤ 𝑦 then 𝑇(𝑥, 𝑦) ≤ 𝑇(𝑥, 1) = 𝑥 = 𝑇(𝑥, 𝑥) ≤ 𝑇(𝑥, 𝑦) such that 𝑇(𝑥, 𝑦) = 𝑥. If 𝑥 ≥ 𝑦 then 𝑇(𝑥, 𝑦) ≥ 𝑇(𝑦, 𝑦) = 𝑦 = 𝑇(1, 𝑦) ≥ 𝑇(𝑥, 𝑦) such that 𝑇(𝑥, 𝑦) = 𝑦. Therefore, we have 𝑇(𝑥, 𝑦) = min{𝑥, 𝑦}. (⇐) Given that 𝑇(𝑥, 𝑦) = min{𝑥, 𝑦} for all 𝑥, 𝑦 ∈ [0,1]. Let 𝑥 = 𝑦, we have 𝑇(𝑥, 𝑥) = min{𝑥, 𝑥} = 𝑥. Therefore, min{𝑥, 𝑦} is idempotent t-norm. ◼ Lemma 2. 𝐶 is a idempotent t-norm if and only if 𝐶(𝑥, 𝑦) = max{𝑥, 𝑦} for all 𝑥, 𝑦 ∈ [0,1]. Proof. (⇒) Take any 𝑥, 𝑦 ∈ [0,1]. If 𝑥 ≥ 𝑦 then 𝐶(𝑥, 𝑦) ≥ 𝐶(𝑥, 0) = 𝑥 = 𝐶(𝑥, 𝑥) ≤ 𝐶(𝑥, 𝑦) such that 𝐶(𝑥, 𝑦) = 𝑥. If 𝑥 ≤ 𝑦 then 𝐶(𝑥, 𝑦) ≤ 𝐶(𝑦, 𝑦) = 𝑦 = 𝐶(0, 𝑦) ≤ 𝐶(𝑥, 𝑦) such that 𝐶(𝑥, 𝑦) = 𝑦. Therefore, we have 𝐶(𝑥, 𝑦) = max{𝑥, 𝑦}. (⇐) Given that 𝐶(𝑥, 𝑦) = max{𝑥, 𝑦} for all 𝑥, 𝑦 ∈ [0,1]. Let 𝑥 = 𝑦, we have 𝑇(𝑥, 𝑥) = max{𝑥, 𝑥} = 𝑥. Therefore, max{𝑥, 𝑦} is idempotent t-conorm. ◼ Corollary 5. If 𝑀 be a 𝑅-module, 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀), 𝑇 and 𝐶 are idempotent t-norm and t- conorm respectively then 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀(𝑀). (IAFM denote intuitionistic anti fuzzy module in [6]). Proof. According to Lemma 1 and 2, if 𝑇 and 𝐶 are idempotent t-norm and t-conorm respectively then 𝑇(𝑥, 𝑦) = min{𝑥, 𝑦} and 𝐶(𝑥, 𝑦) = max{𝑥, 𝑦} for all 𝑥, 𝑦 ∈ [0,1]. Therefore, 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀(𝑀). ◼ Remark 1. Every intuitionistic anti fuzzy module with respect to idempotent t-norm and idempotent t-conorm is intuitionistic anti fuzzy module. The next main result is about some properties with respect to intuitionistic anti fuzzy module t-norm and t-conorm. Theorem 1. If 𝑀 be a 𝑅-module, 𝑎 is a unit on 𝑅, and 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀) then 𝜇𝐴(𝑎𝑥) = 𝜇𝐴(𝑥) and 𝜈𝐴(𝑎𝑥) = 𝜈𝐴(𝑥) for all 𝑥 ∈ 𝑀. The Properties of Intuitionistic Anti Fuzzy Module t-norm and t-conorm Ongky Denny Wijaya 214 Proof. Take any 𝑥 ∈ 𝑀 and let 𝑎 is a unit on 𝑅. Since 𝑎 is a unit, then there exist 𝑏 ∈ 𝑅 such that 𝑎𝑏 = 𝑏𝑎 = 1. Consider that 𝜇𝐴(𝑎𝑥) ≤ 𝜇𝐴(𝑥) = 𝜇𝐴(1𝑥) = 𝜇𝐴((𝑎𝑏)𝑥) = 𝜇𝐴((𝑏𝑎)𝑥) = 𝜇𝐴(𝑏(𝑎𝑥)) ≤ 𝜇𝐴(𝑎𝑥) and 𝜈𝐴(𝑎𝑥) ≥ 𝜈𝐴(𝑥) = 𝜈𝐴(1𝑥) = 𝜈𝐴((𝑎𝑏)𝑥) = 𝜈𝐴((𝑏𝑎)𝑥) = 𝜈𝐴(𝑏(𝑎𝑥)) ≥ 𝜈𝐴(𝑎𝑥). Therefore, 𝜇𝐴(𝑎𝑥) = 𝜇𝐴(𝑥) and 𝜈𝐴(𝑎𝑥) = 𝜈𝐴(𝑥) for all 𝑥 ∈ 𝑀. ◼ Theorem 2. Let 𝑀 be a 𝑅-module, 𝑁 is a submodule of 𝑀, and 𝐴 = (𝜇𝐴, 𝜈𝐴) with degree of membership 𝜇𝐴 and degree of non membership 𝜈𝐴 which defined as 𝜇𝐴(𝑥) = { 1 if 𝑥 ∈ 𝑁 0 if 𝑥 ∉ 𝑁 and 𝜈𝐴(𝑥) = { 1 if 𝑥 ∈ 𝑁 0 if 𝑥 ∉ 𝑁 . If 𝐴′ = (𝜇𝐴𝐶 , 𝜈𝐴) is intuitionistic fuzzy set, where 𝜇𝐴𝐶 : 𝑀 → {0,1} is defined by 𝜇𝐴𝐶 (𝑥) = { 1 if 𝑥 ∉ 𝑁 0 if 𝑥 ∈ 𝑁 then 𝐴′ ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀). Proof. Take any 𝑥, 𝑦 ∈ 𝑀 and 𝑎 ∈ 𝑅. The proof is divided into three cases as follows. (1) First case. If 𝑥, 𝑦 ∈ 𝑁 then 𝑥 + 𝑦 ∈ 𝑁 and 𝑎𝑥 ∈ 𝑁 such that (a) 𝜇𝐴𝐶 (0) = 0. (b) 𝜇𝐴𝐶 (𝑥 + 𝑦) = 0 ≤ 0 = 𝐶(0,0) = 𝐶 (𝜇𝐴𝐶 (𝑥), 𝜇𝐴𝐶 (𝑦)). (c) 𝜇𝐴𝐶 (𝑎𝑥) = 0 ≤ 0 = 𝜇𝐴𝐶 (𝑥). (d) 𝜈𝐴(0) = 1. (e) 𝜈𝐴(𝑥 + 𝑦) = 1 ≥ 1 = 𝑇(1,1) = 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(𝑦)). (f) 𝜈𝐴(𝑎𝑥) = 1 ≥ 1 = 𝜈𝐴(𝑥). (2) Second case. If 𝑥 ∉ 𝑁 and 𝑦 ∈ 𝑁 then 𝑥 + 𝑦 ∉ 𝑁 and 𝑎𝑥 ∉ 𝑁 such that (a) 𝜇𝐴𝐶 (0) = 0. (b) 𝜇𝐴𝐶 (𝑥 + 𝑦) = 1 ≤ 1 = 𝐶(1,0) = 𝐶 (𝜇𝐴𝐶 (𝑥), 𝜇𝐴𝐶 (𝑦)). (c) 𝜇𝐴𝐶 (𝑎𝑥) = 1 ≤ 1 = 𝜇𝐴𝐶 (𝑥). (d) 𝜈𝐴(0) = 1. (e) 𝜈𝐴(𝑥 + 𝑦) = 0 ≥ 0 = 𝑇(0,1) = 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(𝑦)). (f) 𝜈𝐴(𝑎𝑥) = 0 ≥ 0 = 𝜈𝐴(𝑥). (3) Third case. 𝑥, 𝑦 ∉ 𝑁 then 𝑥 + 𝑦 ∉ 𝑁 or 𝑥 + 𝑦 ∈ 𝑁 and 𝑎𝑥 ∉ 𝑁 such that (a) 𝜇𝐴𝐶 (0) = 0. (b) 𝜇𝐴𝐶 (𝑥 + 𝑦) ≤ 1 = 𝐶(1,1) = 𝐶 (𝜇𝐴𝐶 (𝑥), 𝜇𝐴𝐶 (𝑦)). (c) 𝜇𝐴𝐶 (𝑎𝑥) = 1 ≤ 1 = 𝜇𝐴𝐶 (𝑥). (d) 𝜈𝐴(0) = 1. (e) 𝜈𝐴(𝑥 + 𝑦) ≥ 0 = 𝑇(0,0) = 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(𝑦)). (f) 𝜈𝐴(𝑎𝑥) = 0 ≥ 0 = 𝜈𝐴(𝑥). Therefore, based on three cases proof above, we conclude that 𝐴′ ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀). ◼ The Properties of Intuitionistic Anti Fuzzy Module t-norm and t-conorm Ongky Denny Wijaya 215 Theorem 3. Let 𝑀 be a 𝑅-module, 𝑁 is a non empty set of 𝑀, and 𝐴 = (𝜇𝐴, 𝜈𝐴) is an intuitionistic fuzzy set which degree of membership 𝜇𝐴: 𝑁 → [0,1] and degree of non membership 𝜈𝐴: 𝑁 → [0,1] defined as 𝜇𝐴(𝑥) = { 0 if 𝑥 ∈ 𝑁 𝑐1 if 𝑥 ∉ 𝑁 and 𝜈𝐴(𝑥) = { 1 if 𝑥 ∈ 𝑁 𝑐2 if 𝑥 ∉ 𝑁 , for 0 ≤ 𝑐1 ≤ 1, 0 ≤ 𝑐2 ≤ 1, and , 0 ≤ 𝑐1 + 𝑐2 ≤ 1. 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀) if and only if 𝑁 is a submodule of 𝑀. Proof. (⇒) Take any 𝑥, 𝑦 ∈ 𝑀 and 𝑎 ∈ 𝑅 which means 𝜇𝐴(𝑥) = 𝜇𝐴(𝑦) = 0 and 𝜈𝐴(𝑥) = 𝜈𝐴(𝑦) = 1. Based on given 𝜇𝐴(𝑥) and 𝜈𝐴(𝑥), we have 𝜇𝐴(0) = 0 and 𝜈𝐴(0) = 1 imply 0 ∈ 𝑁. Now, consider that 𝜇𝐴(𝑥 − 𝑦) ≤ 𝐶(𝜇𝐴(𝑥), 𝜇𝐴(−𝑦)) = 𝐶(𝜇𝐴(𝑥), 𝜇𝐴(𝑦)) = 𝐶(0,0) = 0 and 𝜈𝐴(𝑥 − 𝑦) ≥ 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(−𝑦)) = 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(𝑦)) = 𝑇(1,1) = 1, which mean 𝑥 − 𝑦 ∈ 𝑁. Consider that 𝜇𝐴(𝑎𝑥) ≤ 𝜇𝐴(𝑥) = 0 and 𝜈𝐴(𝑎𝑥) ≥ 𝜈𝐴(𝑥) = 1, which mean 𝑎𝑥 ∈ 𝑁. Based on Definition 2, 𝑁 is a submodule of 𝑀. (⇐) The proof of 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀) directly from Definition 10 as follows. Take any 𝑥, 𝑦 ∈ 𝑀 and 𝑎 ∈ 𝑅. (1) Obviously, 0 ∈ 𝑁 imply 𝜇𝐴(0) = 0. (2) The proof is divided into three cases as follows. (a) If 𝑥 ∈ 𝑁 and 𝑦 ∈ 𝑁 then 𝑥 + 𝑦 ∈ 𝑁 and 𝜇𝐴(𝑥) = 𝜇𝐴(𝑦) = 0. Thus, 𝜇𝐴(𝑥 + 𝑦) = 0 ≤ 0 = 𝐶(0,0) = 𝐶(𝜇𝐴(𝑥), 𝜇𝐴(𝑦)). (b) If 𝑥 ∈ 𝑁 and 𝑦 ∉ 𝑁 then 𝑥 + 𝑦 ∉ 𝑁, 𝜇𝐴(𝑥) = 0, dan 𝜇𝐴(𝑦) = 𝑐1. Thus, 𝜇𝐴(𝑥 + 𝑦) = 𝑐1 ≤ 𝑐1 = 𝐶(0, 𝑐1) = 𝐶(𝜇𝐴(𝑥), 𝜇𝐴(𝑦)). (c) If 𝑥 ∉ 𝑁 and 𝑦 ∉ 𝑁 then 𝑥 + 𝑦 ∈ 𝑁 or 𝑥 + 𝑦 ∉ 𝑁, 𝜇𝐴(𝑥) = 𝑐1, and 𝜇𝐴(𝑦) = 𝑐1. If 𝑥 + 𝑦 ∈ 𝑁 then 𝜇𝐴(𝑥 + 𝑦) = 0 = 𝐶(0,0) ≤ 𝐶(𝑐1, 𝑐1) = 𝐶(𝜇𝐴(𝑥), 𝜇𝐴(𝑦)). If 𝑥 + 𝑦 ∉ 𝑁 then 𝜇𝐴(𝑥 + 𝑦) = 𝑐1 = 𝐶(𝑐1, 0) ≤ 𝐶(𝑐1, 𝑐1) = 𝐶(𝜇𝐴(𝑥), 𝜇𝐴(𝑦)). (3) The proof is divided into two cases as follows. (a) If 𝑥 ∈ 𝑁 then 𝜇𝐴(𝑥) = 0 and 𝑎𝑥 ∈ 𝑁. Thus, 𝜇𝐴(𝑎𝑥) = 0 ≤ 0 = 𝜇𝐴(𝑥). (b) If 𝑥 ∉ 𝑁 then 𝜇𝐴(𝑥) = 𝑐1 and 𝑎𝑥 ∈ 𝑁 or 𝑎𝑥 ∉ 𝑁. If 𝑎𝑥 ∈ 𝑁 then 𝜇𝐴(𝑎𝑥) = 0 ≤ 𝑐1 = 𝜇𝐴(𝑥). If 𝑎𝑥 ∉ 𝑁 then 𝜇𝐴(𝑎𝑥) = 𝑐1 ≤ 𝑐1 = 𝜇𝐴(𝑥). (4) Obviously, 0 ∈ 𝑁 imply 𝜈𝐴(0) = 1. (5) The proof is divided into three cases as follows. (a) If 𝑥 ∈ 𝑁 and 𝑦 ∈ 𝑁 then 𝑥 + 𝑦 ∈ 𝑁 and 𝜈𝐴(𝑥) = 𝜈𝐴(𝑦) = 1. Thus, 𝜈𝐴(𝑥 + 𝑦) = 1 ≥ 1 = 𝑇(1,1) = 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(𝑦)). (b) If 𝑥 ∈ 𝑁 and 𝑦 ∉ 𝑁 then 𝑥 + 𝑦 ∉ 𝑁, 𝜈𝐴(𝑥) = 1, dan 𝜈𝐴(𝑦) = 𝑐2. Thus, 𝜈𝐴(𝑥 + 𝑦) = 𝑐2 ≥ 𝑐2 = 𝑇(1, 𝑐2) = 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(𝑦)). The Properties of Intuitionistic Anti Fuzzy Module t-norm and t-conorm Ongky Denny Wijaya 216 (c) If 𝑥 ∉ 𝑁 and 𝑦 ∉ 𝑁 then 𝑥 + 𝑦 ∈ 𝑁 or 𝑥 + 𝑦 ∉ 𝑁, 𝜈𝐴(𝑥) = 𝑐2, and 𝜈𝐴(𝑦) = 𝑐2. If 𝑥 + 𝑦 ∈ 𝑁 then 𝜈𝐴(𝑥 + 𝑦) = 1 = 𝑇(1,1) ≥ 𝑇(𝑐2, 𝑐2) = 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(𝑦)). If 𝑥 + 𝑦 ∉ 𝑁 then 𝜈𝐴(𝑥 + 𝑦) = 𝑐2 = 𝑇(𝑐2, 1) ≥ 𝑇(𝑐2, 𝑐2) = 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(𝑦)). (6) The proof is divided into two cases as follows. (a) If 𝑥 ∈ 𝑁 then 𝜈𝐴(𝑥) = 1 and 𝑎𝑥 ∈ 𝑁. Thus, 𝜈𝐴(𝑎𝑥) = 1 ≥ 1 = 𝜈𝐴(𝑥). (b) If 𝑥 ∉ 𝑁 then 𝜈𝐴(𝑥) = 𝑐2 and 𝑎𝑥 ∈ 𝑁 or 𝑎𝑥 ∉ 𝑁. If 𝑎𝑥 ∈ 𝑁 then 𝜈𝐴(𝑎𝑥) = 1 ≥ 𝑐2 = 𝜈𝐴(𝑥). If 𝑎𝑥 ∉ 𝑁 then 𝜈𝐴(𝑎𝑥) = 𝑐2 ≥ 𝑐2 = 𝜈𝐴(𝑥). Therefore, 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀). ◼ Theorem 4. If 𝑀 be a 𝑅-module and 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀) then the set 𝑁 = {𝑥 ∈ 𝑀|𝐴(𝑥) = (0,1)} is a submodule of 𝑀. Proof. Take any 𝑥, 𝑦 ∈ 𝑁 and 𝑎 ∈ 𝑅. This means 𝜇𝐴(𝑥) = 𝜇𝐴(𝑦) = 0 and 𝜈𝐴(𝑥) = 𝜈𝐴(𝑦) = 1. Now, we want to show 𝑁 is a submodule of 𝑀 as follows. (1) Consider that 𝜇𝐴(𝑥 − 𝑦) ≤ 𝐶(𝜇𝐴(𝑥), 𝜇𝐴(−𝑦)) ≤ 𝐶(𝜇𝐴(𝑥), 𝜇𝐴(𝑦)) = 𝐶(0,0) = 0, 𝜈𝐴(𝑥 − 𝑦) ≥ 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(−𝑦)) ≥ 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(𝑦)) = 𝑇(1,1) = 1. Now, we have 𝜇𝐴(𝑥 − 𝑦) ≤ 0 and 𝜈𝐴(𝑥 − 𝑦) ≥ 1 which means 𝜇𝐴(𝑥 − 𝑦) = 0 and 𝜈𝐴(𝑥 − 𝑦) = 1. Thus, 𝐴(𝑥 − 𝑦) = (𝜇𝐴(𝑥 − 𝑦), 𝜈𝐴(𝑥 − 𝑦)) = (0,1) which imply 𝑥 − 𝑦 ∈ 𝑁. (2) Consider that 𝜇𝐴(𝑎𝑥) ≤ 𝜇𝐴(𝑥) = 0 and 𝜈𝐴(𝑎𝑥) ≥ 𝜈𝐴(𝑥) = 1 which 𝜇𝐴(𝑎𝑥) = 0 and 𝜈𝐴(𝑎𝑥) = 1 must hold. Thus, 𝐴(𝑎𝑥) = (𝜇𝐴(𝑎𝑥), 𝜈𝐴(𝑎𝑥)) = (0,1), which implies 𝑎𝑥 ∈ 𝑁. Therefore, 𝑁 is a submodule of 𝑀. ◼ Theorem 5. Let 𝑀 be a 𝑅-module and 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀). If 𝐴(𝑥 − 𝑦) = (0,1) then A(x) = 𝐴(𝑦) for all 𝑥, 𝑦 ∈ 𝑀. Proof. Take any 𝑥, 𝑦 ∈ 𝑀. Given 𝜇𝐴(𝑥 − 𝑦) = 0 and 𝜈𝐴(𝑥 − 𝑦) = 1. We will show 𝜇𝐴(𝑥) = 𝜇𝐴(𝑦) and 𝜈𝐴(𝑥) = 𝜈𝐴(𝑦) directly from axiom on the Definition 10. Consider that 𝜇𝐴(𝑥) = 𝜇𝐴(𝑥 − 𝑦 + 𝑦) ≤ 𝐶(𝜇𝐴(𝑥 − 𝑦), 𝜇𝐴(𝑦)) = 𝐶(0, 𝜇𝐴(𝑦)) = 𝜇𝐴(𝑦) and The Properties of Intuitionistic Anti Fuzzy Module t-norm and t-conorm Ongky Denny Wijaya 217 𝜇𝐴(𝑦) = 𝜇𝐴(𝑥 − 𝑥 + 𝑦) = 𝜇𝐴(𝑥 − (𝑥 − 𝑦)) ≤ 𝐶 (𝜇𝐴(𝑥), 𝜇𝐴(−(𝑥 − 𝑦))) = 𝐶(𝜇𝐴(𝑥), 𝜇𝐴(𝑥 − 𝑦)) = 𝐶(𝜇𝐴(𝑥), 0) = 𝜇𝐴(𝑥). Thus, we have 𝜇𝐴(𝑥) ≤ 𝜇𝐴(𝑦) and 𝜇𝐴(𝑦) ≤ 𝜇𝐴(𝑥) or equivalently 𝜇𝐴(𝑥) = 𝜇𝐴(𝑦). Now, consider that 𝜈𝐴(𝑥) = 𝜈𝐴(𝑥 − 𝑦 + 𝑦) ≥ 𝑇(𝜈𝐴(𝑥 − 𝑦), 𝜈𝐴(𝑦)) = 𝑇(1, 𝜈𝐴(𝑦)) = 𝜈𝐴(𝑦) and 𝜈𝐴(𝑦) = 𝜈𝐴(𝑥 − 𝑥 + 𝑦) = 𝜈𝐴(𝑥 − (𝑥 − 𝑦)) ≥ 𝑇 (𝜈𝐴(𝑥), 𝜈𝐴(−(𝑥 − 𝑦))) = 𝑇(𝜈𝐴(𝑥), 𝜈𝐴(𝑥 − 𝑦)) = 𝑇(𝜈𝐴(𝑥), 1) = 𝜈𝐴(𝑥). Thus, we have 𝜈𝐴(𝑥) ≤ 𝜈𝐴(𝑦) and 𝜈𝐴(𝑦) ≤ 𝜈𝐴(𝑥) or equivalently 𝜈𝐴(𝑥) = 𝜈𝐴(𝑦). Therefore, 𝐴(𝑥) = (𝜇𝐴(𝑥), 𝜈𝐴(𝑥)) = (𝜇𝐴(𝑦), 𝜈𝐴(𝑦)) = 𝐴(𝑦). ◼ In the next theorem, we discuss about properties of intuitionistic anti fuzzy module t-norm and t-conorm with respect to module homomorphism, image, pre-image, and anti-image. Theorem 6. Let 𝑀 and 𝑁 be a 𝑅-module, 𝛼: 𝑀 → 𝑁 is 𝑅-module homomorphism, and 𝐵 = (𝜇𝐵 , 𝜈𝐵 ) is intuitionistic fuzzy set on 𝑁. If 𝐵 = (𝜇𝐵 , 𝜈𝐵 ) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑁) then 𝛼 −1(𝐵) = (𝜇𝛼−1(𝐵), 𝜈𝛼−1(𝐵)) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀). Proof. Let identity element of addition on 𝑀 and 𝑁 is denoted by 0𝑀 and 0𝑁 respectively. Given 𝐵 = (𝜇𝐵 , 𝜈𝐵 ) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑁). We will prove 𝛼 −1(𝐵) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀). Take any 𝑥1, 𝑥2 ∈ 𝑀 and 𝑎 ∈ 𝑅. Consider that (1) 𝜇𝛼−1(𝐵)(0𝑀 ) = 𝜇𝐵 (𝛼(0𝑀)) = 𝜇𝐵 (0𝑁 ) = 0. (2) 𝜇𝛼−1(𝐵)(𝑥1 + 𝑥2) = 𝜇𝐵 (𝛼(𝑥1 + 𝑥2)) = 𝜇𝐵 (𝛼(𝑥1) + 𝛼(𝑥2)) ≤ 𝐶 (𝜇𝐵 (𝛼(𝑥1)), 𝜇𝐵 (𝛼(𝑥2))) = 𝐶 (𝜇𝛼−1(𝐵)(𝑥1), 𝜇𝛼−1(𝐵)(𝑥2)). (3) 𝜇𝛼−1(𝐵)(𝑎𝑥1) = 𝜇𝐵 (𝛼(𝑎𝑥1)) = 𝜇𝐵 (𝑎𝛼(𝑥1)) ≤ 𝜇𝐵 (𝛼(𝑥1)) = 𝜇𝛼−1(𝐵)(𝑥1). (4) 𝜈𝛼−1(𝐵)(0𝑀) = 𝜈𝐵 (𝛼(0𝑀)) = 𝜈𝐵 (0𝑁 ) = 1. (5) 𝜈𝛼−1(𝐵)(𝑥1 + 𝑥2) = 𝜈𝐵 (𝛼(𝑥1 + 𝑥2)) = 𝜈𝐵 (𝛼(𝑥1) + 𝛼(𝑥2)) ≥ 𝑇 (𝜈𝐵 (𝛼(𝑥1)), 𝜈𝐵 (𝛼(𝑥2))) = 𝑇 (𝜈𝛼−1(𝐵)(𝑥1), 𝜈𝛼−1(𝐵)(𝑥2)). (6) 𝜈𝛼−1(𝐵)(𝑎𝑥1) = 𝜈𝐵 (𝛼(𝑎𝑥1)) = 𝜈𝐵 (𝑎𝛼(𝑥1)) ≥ 𝜈𝐵 (𝛼(𝑥1)) = 𝜈𝛼−1(𝐵)(𝑥1). Therefore, 𝛼−1(𝐵) = (𝜇𝛼−1(𝐵), 𝜈𝛼−1(𝐵)) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀). ◼ Theorem 7. Let 𝑀 and 𝑁 be a 𝑅-module, 𝛼: 𝑀 → 𝑁 is 𝑅-module epimorphism, and 𝐴 = (𝜇𝐴, 𝜈𝐴) is intuitionistic fuzzy set on 𝑀. If 𝐴 = (𝜇𝐴, 𝜈𝐴) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑀) then �̂�(𝐴) = (𝜇�̂�(𝐴), 𝜈�̂�(𝐴)) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑁). The Properties of Intuitionistic Anti Fuzzy Module t-norm and t-conorm Ongky Denny Wijaya 218 Proof. Take any 𝑦1, 𝑦2 ∈ 𝑁 and 𝑎 ∈ 𝑅. Given 𝛼 is 𝑅-module epimorphism, i.e. surjective homomorphism, obviously 𝛼−1(𝑦1 + 𝑦2) ≠ ∅. Let 𝑦1 = 𝛼(𝑥1) and 𝑦2 = 𝛼(𝑥2), for all 𝑥1, 𝑥2 ∈ 𝑀. Consider that (1) 𝜇�̂�(𝐴)(0𝑁 ) = min 𝑥∈𝛼−1(0𝑁) 𝜇𝐴(𝑥) = 𝜇𝐴(0𝑀) = 0. (2) 𝜇�̂�(𝐴)(𝑦1 + 𝑦2) = 𝜇�̂�(𝐴)(𝛼(𝑥1) + 𝛼(𝑥2)) = 𝜇�̂�(𝐴)(𝛼(𝑥1 + 𝑥2)) = min 𝑥∈𝛼−1(𝛼(𝑥1+𝑥2)) 𝜇𝐴(𝑥) = 𝜇𝐴(𝑥1 + 𝑥2) ≤ 𝐶(𝜇𝐴(𝑥1), 𝜇𝐴(𝑥2)) = 𝐶 ( min 𝑥∈𝛼−1(𝛼(𝑥1)) 𝜇𝐴(𝑥), min 𝑥∈𝛼−1(𝛼(𝑥2)) 𝜇𝐴(𝑥) ) = 𝐶 (𝜇�̂�(𝐴)(𝛼(𝑥1)), 𝜇�̂�(𝐴)(𝛼(𝑥2))) = 𝐶 (𝜇�̂�(𝐴)(𝑦1), 𝜇�̂�(𝐴)(𝑦2)). (3) 𝜇�̂�(𝐴)(𝑎𝑦1) = 𝜇�̂�(𝐴)(𝑎𝛼(𝑥1)) = 𝜇�̂�(𝐴)(𝛼(𝑎𝑥1)) = min 𝑥∈𝛼−1(𝛼(𝑎𝑥1)) 𝜇𝐴(𝑥) = 𝜇𝐴(𝑎𝑥1) ≤ 𝜇𝐴(𝑥1) = min 𝑥∈𝛼−1(𝛼(𝑥1)) 𝜇𝐴(𝑥) = 𝜇�̂�(𝐴)(𝛼(𝑥1)) = 𝜇�̂�(𝐴)(𝑦1). (4) 𝜈�̂�(𝐴)(0𝑁 ) = max 𝑥∈𝛼−1(0𝑁) 𝜈𝐴(𝑥) = 𝜈𝐴(0𝑀) = 1. (5) 𝜈�̂�(𝐴)(𝑦1 + 𝑦2) = 𝜈�̂�(𝐴)(𝛼(𝑥1) + 𝛼(𝑥2)) = 𝜈�̂�(𝐴)(𝛼(𝑥1 + 𝑥2)) = max 𝑥∈𝛼−1(𝛼(𝑥1+𝑥2)) 𝜈𝐴(𝑥) = 𝜈𝐴(𝑥1 + 𝑥2) ≥ 𝑇(𝜈𝐴(𝑥1), 𝜈𝐴(𝑥2)) = 𝑇 ( max 𝑥∈𝛼−1(𝛼(𝑥1)) 𝜈𝐴(𝑥), max 𝑥∈𝛼−1(𝛼(𝑥2)) 𝜈𝐴(𝑥) ) = 𝑇 (𝜈�̂�(𝐴)(𝛼(𝑥1)), 𝜈�̂�(𝐴)(𝛼(𝑥2))) = 𝑇 (𝜈�̂�(𝐴)(𝑦1), 𝜈�̂�(𝐴)(𝑦2)). (6) 𝜈�̂�(𝐴)(𝑎𝑦1) = 𝜈�̂�(𝐴)(𝑎𝛼(𝑥1)) = 𝜈�̂�(𝐴)(𝛼(𝑎𝑥1)) = max 𝑥∈𝛼−1(𝛼(𝑎𝑥1)) 𝜈𝐴(𝑥) = 𝜈𝐴(𝑎𝑥1) ≥ 𝜈𝐴(𝑥1) = max 𝑥∈𝛼−1(𝛼(𝑥1)) 𝜈𝐴(𝑥) = 𝜈�̂�(𝐴)(𝛼(𝑥1)) = 𝜈�̂�(𝐴)(𝑦1). Therefore, �̂�(𝐴) = (𝜇�̂�(𝐴), 𝜈�̂�(𝐴)) ∈ 𝐼𝐴𝐹𝑀𝑇𝐶(𝑁). ◼ CONCLUSIONS We have define the intuitionistic anti fuzzy module t-norm and t-conorm and investigate their general properties and properties with respect to module homomorphism, maps, pre-image, and anti-image from intuitionistic fuzzy sets. We give some example, corollary, and the properties. 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