NEW FORM OF SOFT SETS AND ITS PROPERTIES Basavaraj M Ittanagi1, Prakash R2∗ and Chetana C3 1Department of Mathematics, Siddaganga Institute of Technology, Tumakuru-572103, Karnakata State, India. E-mail: dr.basavarajsit@gmail.com 2∗Department of Mathematics, RV College of Engineering, Mysore Road, Bengaluru–560 059, Karnakata State, India. E-mail: prakashr@rvce.edu.in 3Department of Mathematics, Sri Siddhartha Institute of Technology, SSAHE,Tumakuru-572105, Karnakata State, India. E-mail: chetanachandran@gmail.com Abstract: Molodtsov introduced the concept of soft set theory as a general mathematical tool for dealing with uncertainty. Many researchers have studied this theory and developed several models to solve decision-making and medical diagnostic problems, but most of these models deal only one set of parameters. This causes problems for users, especially with those who use questionnaires in their work and studies. Also Alkhazaleh and Salleh, also introduced the con- cept of soft-expert sets. This structure can be considered as a generalization of soft-sets in which experts and their opinions have been added to make deci- sion analysis easier to handle. In our model, is more generalization of soft- set and soft-expert set, the collection of more specific information about object sets using mappings. This concept is more powerful for information tables, since collection of the information is very particular to define by mapping and also this model is approaches to rough set theory and information system. Keywords: Soft sets, soft expert sets and swarm sets. 1. Introduction Most of the problems in engineering, medicine, economics, environment, etc. are related to various uncertainties. Molodtsov [1] initiated the concept of soft- set theory as a mathematical tool for dealing with uncertainty. After the work of Molodtsov, some operations and applications of the soft-set were introduced by Chen et al. [2] and Maji et al. [3, 4]. Alkhazaleh et al. [5] introduced the concept of soft-multi set as a generalization of soft-set. They also defined the concepts of possible fuzzy soft-sets and fuzzy parameterized interval valued fuzzy soft-sets in [6, 7] and gave their application to decision making and med- ical diagnosis. Alkhazaleh and Salleh [9] introduced the concept of soft-expert sets. Many researchers have studied this theory and developed several models to solve decision-making and medical diagnostic problems, but most of these models deal only with one set of parameters, and if we want to take the number of attributes and sets of parameters, perform some operations such as union, intersection, and so on. This causes problems among users, especially with those who use questionnaires in their work and studies. In our model the user can know the set of parameters in one model without any intervention. Even after any operation on our model the user can know the number of attributes and sets of parameters. So in this paper we introduce the concept of swarm set deals with associate parameters in each of its attributes of objects, which will be more Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) https://internationalpubls.com 91 Received: 05-08-2024, Revised: 22-09-2024, Accepted: 09-10-2024 effective and useful. We also introduced basic operations namely, complement, union and intersection. Finally, we give an application on real estate problem to illustrate the concept is discussed. 2. Preliminaries: Basic Definitions Revisited In this section, we recall some basic notions in soft set theory. Molodtsov [1] defined soft set in the following way. Let U be an initial universe set and let E be a set of parameters. Let P (U) denote the power set of U and A ⊆ E. Definition 2.1: A pair (F,A) is called a soft set over U , where F is a mapping defined by F : A → P (U). In other words, a soft set over U is a parameterized family of subsets of the universe U . For ϵ ∈ A,F (ϵ) may be considered as the set of ϵ-approximate elements of the soft set (F,A). Definition 2.2: For two soft sets (F,A) and (G,B) over a common universe U , we say that (F,A) is a soft subset of (G,B) if (i) A ⊂ B and (ii) ∀ ϵ ∈ A, F (ϵ) and G(ϵ) are identical approximations. Write (F,A)⊂̃(G,B). (F,A) is said to be a soft super set of (G,B), if (G,B) is a soft subset of (F,A). Denoted by (F,A)⊃̃(G,B). Definition 2.3: For two soft sets (F,A) and (G,B) over a common universe U are said to be soft equal if (F,A) is a soft subset of (G,B) and (G,B) is a soft subset of (F,A). Definition 2.4: Let E = {e1, e2, ..., en} be a set of parameters. The NOT set of E denoted by ¬E and is defined by ¬E = {¬e1,¬e2, ...,¬en} where ¬ei = not ei, ∀ i. Definition 2.5: The complement of a soft set (F,A) is denoted by (F,A)C and is defined by (F,A)C = (FC ,¬A), where FC : ¬A → P (U) is a mapping given by FC(α) = U − F (¬α), ∀ α ∈ ¬A. Definition 2.6: A soft set (F,A) over U is said to be a NULL soft set denoted by ϕ if for all ϵ ∈ A,F (ϵ) = ϕ(null set). Definition 2.7: A soft set (F,A) over U is said to be an absolute soft set denoted by à if for all ϵ ∈ A,F (ϵ) = U . Definition 2.8: If (F,A) and (G,B) are two soft sets, then (F,A) AND (G,B) denoted by (F,A) ∧ (G,B) is defined by (F,A) ∧ (G,B) = (H,A × B), where H(α, β) = F (α) ∩ F (β), for all (α, β) ∈ A×B. Definition 2.9: If (F,A) and (G,B) are two soft sets, then (F,A) OR (G,B) denoted by (F,A) ∨ (G,B) is defined by (F,A) ∨ (G,B) = (O,A × B), where O(α, β) = F (α) ∪ F (β), for all (α, β) ∈ A×B. Definition 2.10: The union of two soft sets (F,A) and (G,B) over a common universe U is the soft set (H,C), where C = A ∪B and ∀ e ∈ C, H(e) =  F (e), if e ∈ A−B G(e), if e ∈ B −A F (e) ∪G(e), if e ∈ A ∩B (1) Write (F,A)∪̃(G,B) = (H,C) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) https://internationalpubls.com 92 Definition 2.11: The intersection (H,C) of two soft sets (F,A) and (G,B) over a common universe U , denoted (F,A) ∩ (G,B), is defined as C = A ∩ B, and H(e) = F (e) ∩G(e) for all e ∈ C Definition 2.12: The extended intersection of two soft sets (F,A) and (G,B) over a common universe U is the soft set (H,C), where C = A∪B and ∀ e ∈ C, H(e) =  F (e), if e ∈ A−B G(e), if e ∈ B −A F (e) ∩G(e), if e ∈ A ∩B (2) 3. Swarm Sets Let U be the set of objects, A be a finite set of attributes in which each attribute is a word or sentence and V is a collection of all associated parameter set with each attribute, such that V = ⋃ Va, ∀ a ∈ A. For example, consider the attribute set A = {gender, age}, then the asso- ciate parameter set of an attribute gender is Vgender = {male, female} and age is Vage = {young, adult, old}. The collection of all associate parameter set is V = ⋃ a∈AVa = Vgender ∪ Vage = {male, female, young, adult, old}. Definition 3.1: The function fA is called swarm map defined by fA : V → P (U), where P (U) denotes power set of object set U and V = ⋃ a∈AVa. In other words, a swarm map fA is associate parameters of each of its at- tributes of objects. Definition 3.2: A pair (fA, V ) is called swarm set over the swarm map fA : V → P (U), where P (U) denotes power set of U and V = ⋃ a∈AVa. Example 3.3: Consider U = {u1, u2, u3} is the set of objects and A = {colour, shape, size, category} is the set of attributes. The associate parameters set of an attribute colour is Vcolour = {orange, yellow, red}, shape is Vshape = {round, long}, size is Vsize = {small,medium} and category is Vcategory = {apple, banana, orange}. Tabular representation of swarm set is in Table 1. Table 1: Objects Attributes U colour shape size category u1 red round medium apple u2 yellow long medium banana u3 orange round small orange The collection of all associate parameter set is V = Vcolour ∪ Vshape ∪ Vsize ∪ Vcategory = {orange, yellow, red, round, long, small, medium, apple, banana, orange}. The swarm map fA : V → P (U) is defined by, fA(orange) = {u3} or fcolour(orange) = {u3}, fcolour(yellow) = {u2}, fcolour(red) = {u1}, fshape(round) = {u1, u3}, fshape(long) = {u2}, fsize(small) = {u3}, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) https://internationalpubls.com 93 fsize(medium) = {u1, u2}, fcategory(apple) = {u1}, fcategory(banana) = {u2}, fcategory(orange) = {u3}. Then the swarm set is (fA, V ) = {(u1, {red, round,medium, apple}), (u2, {yellow, long,medium, banana}), (u3, {orange, round, small, orange}). 4. Complete and Incomplete Swarm sets In this section, we introduce the complete and incomplete swarm sets with examples. Definition 4.1: The swarm set (fA, V ) is called complete swarm set if⋃ a∈A(fA(ea)) = U , otherwise incomplete swarm set. Example 4.2: Example 3.3 is complete swarm set. Example 4.3: Consider U = {h1, h2, h3, h4, h5, h6} is set of objects and A = {Type, Price} = {T, P} is set attributes. The associate parameters set of an attribute, Type is VT = {muddy,wooden} = {et1, et2} and Price is VP = {low, high} = {ep1, ep2}. The collection of all associate parameter set is V = VT ∪ VP = {muddy,wooden, low, high} = {et1, et2, ep1, ep2}. The swarm map fA : V → P (U) is defined by fA(et1) = {h1, h3}, fA(et2) = {h2, h6}, fA(ep1) = {h2, h3}, fA(ep2) = {h1, h5, h6}. Then the swarm set is (fA, V ) = {(h1, {et1, ep2}), (h2, {et2, ep1}), (h3, {et1, ep1}), (h4, {ϕ, ϕ}), (h5, {ϕ, ep2}), (h6, {et2, ep2})}. Therefore (fA, V ) is not a complete swarm set. Since fA(et1) ∪ fA(et2) = {h1, h3} ∪ {h2, h6} = {h1, h2, h3, h6} ̸= U . Thus, (fA, V ) is incomplete swarm set. Note that in this paper only complete swarm sets are used. 5. Operations of Swarm Sets. In this section, give definitions of its basic operations with examples. Let (fA, V ) and (fB , V ) are two swarm sets over common object set U and A, B are sets of attributes. The associate parameters set VA = ⋃ a∈A Va and the associate parameters set VB = ⋃ b∈B Vb respectively. The collection of all associate parameter set is V = (∪a∈AVa) ⋃ (∪b∈BVb). Definition 5.1: Swarm subset: The swarm set (fA, V ) is subset of swarm set (fB , V ) if i) A ⊂ B ii) fA(eα) ⊂ fB(eα) , ∀ eα ∈ V where fA(eα) and fB(eα) are identical approximation, write (fA, V )⊂̃(fB , V ), i.e (fA, V ) is swarm subset of (fB , V ). Also (fA, V ) is said to be swarm super set of (fB , V ), if (fB , V ) is swarm subset of (fA, V ). Denote it by (fA, V )⊃̃ (fB , V ). Definition 5.2: Equality of two swarm sets : Swarm sets (fA, V ) and (fB , V ) are said to be equal if (fA, V ) is swarm subset of (fB , V ) and (fB , V ) is swarm subset of (fA, V ). Example 5.3: Consider (fA, V ) and (fB , V ) are two swarm sets over common object set U = {h1, h2, h3, h4, h5, h6} and A = {T, P }, B = {T, F, P, C} are sets of attributes. The associate parameter set of an attribute A is VT = {muddy, wooden, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) https://internationalpubls.com 94 RCC} = {et1, et2, et3} and VP = {low, high, very high} = {ep1, ep2, ep3}. The associate parameter set of an attribute B is VT = {muddy,wooden, RCC} = {et1, et2, et3}, VF = {east, west, north, south} = {ef1, ef2, ef3, ef4}, VP = {low, high, very high} = {ep1, ep2, ep3} and VC = {bad, good} = {ec1, ec2}. The collection of all associate parameter set is V = {VT ∪ VP } ∪ {VT ∪ VF ∪ VP ∪ VC} = {VT ∪ VP ∪ VF ∪ VC} = {muddy,wooden,RCC, low, high, very high, east, west, north, south, bad, good} = {et1, et2, et3, ep1, ep2, ep3, ef1, ef2, ef3, ef4, ec1, ec2}. The swarm map fA : V → P (U) is defined by fA(et1) = {h2, h3}, fA(et2) = {h1, h4}, fA(et3) = {h5, h6}, fA(ep1) = {h2, h4, h5}, fA(ep2) = {h3}, fA(ep3) = {h1, h6}, Then the swarm set is (fA, V ) = {(h1, {et2, ep3}), (h2, {et1, ep1}), (h3, {et1, ep2}), (h4, {et2, ep1}), (h5, {et3, ep1}), (h6, {et3, ep3})}. The swarm map fB : V → P (U) is defined by fB(et1) = {h2, h3}, fB(et2) = {h1, h4}, fB(et3) = {h5, h6}, fB(ef1) = {h2, h3, h5, h6}, fB(ef2) = {ϕ}, fB(ef3) = {h1, h4}, fB(ef4) = {ϕ}, fB(ep1) = {h2, h4, h5}, fB(ep2) = {h3}, fB(ep3) = {h1, h6}, fB(ec1) = {h1, h2, h4, h5}, fB(ec2) = {h3, h6}. Then the swarm set is (fB , V ) = {(h1, {et2, ef3, ep3, ec1}), (h2, {et1, ef1, ep1, ec1}), (h3, {et1, ef1, ep2, ec2}), (h4, {et2, ef3, ep1, ec1}), (h5, {et3, ef1, ep1, ec1}), (h6, {et3, ef1, ep3, ec2})}. Therefore, (fA, V )⊂̃(fB , V ). Definition 5.4: NOT set of attribute and NOT associate parameters set: Let A = {P,Q,R} be the set of attributes. The associate parameter set of an attribute P is VP = {ep1, ep2}, Q is VQ = {eq1, eq2} and R is VR = {er1, er2}. The collection of all associate parameter set is V = VP ∪ VQ ∪ VR = {ep1, ep2, eq1, eq2, er1, er2}. The NOT set of attribute and NOT associate parameters set are denoted by ¬A and ¬V respectively and defined by ¬A = {¬P,¬Q,¬R} and ¬V = (¬VP ) ∪ (¬VQ) ∪ (¬VR) = {¬ep1,¬ep2,¬eq1,¬eq2,¬er1,¬er2}. That is ¬epi = not epi,¬eqi = not eqi,¬eri = not eri, ∀i. Example 5.5: Consider A = {Type, Condition} = {T,C} is set of attributes and The associate parameter set of an attribute, Type is VT = {muddy,wooden, RCC} = {et1, et2, et3}, Condition is VC = {bad, good} = {ec1, ec2}. The collection of all associate parameter set is V = (VT )∪ (VC) = {muddy, wooden,RCC, bad, good} = {et1, et2, et3, ec1, ec2}. NOT set of an attribute ¬A = {¬T,¬C} and NOT associate parameters set of an attribute, Type is ¬VT = {not muddy, not wooden, not RCC} = {¬et1,¬et2,¬et3}, Condition is ¬VC = {not bad, not good} = {¬ec1,¬ec2}. Therefore the collection of all NOT associate parameters set is ¬V = {(¬VT )∪ (¬VC)} = {not muddy, not wooden, not RCC, not bad, not good} = {¬et1,¬et2, ¬et3,¬ec1,¬ec2}. Proposition 5.6: Let (fA, V ) and (fB , V ) are two swarm sets over common s⋃object et U and A, B are⋃sets of attributes. The associate parameters sets are VA = a∈A Va and VB = b∈B V⋃b respectively and the collection of all associate parameter set is V = (∪a∈AVa) (∪b∈BVb). Then i) ¬(¬(A)) = A Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) https://internationalpubls.com 95 ii) ¬(A ∪B) = (¬A ∪ ¬B) iii) ¬(A ∩B) = (¬A ∩ ¬B) Definition 5.7: Complement of swarm set : A complement of swarm set (fA, V ) is denoted by (fA, V )C and is defined by (fA, V )C = (fC ¬A,¬V ). The complement of swarm map fC ¬A : ¬V → P (U) is defined by fC ¬A(¬ea) = U − fA(ea), ∀ a ∈ VA. Clearly, [(fA, V )C ]C = (fA, V ). Example 5.8: Consider U = {h1, h2, h3, h4, h5, h6, h7, h8, h9, h10} and A = {Type, Face, Price,Durable, Condition} = {T, F, P,D,C} . The collection of all associate parameter set is V = {muddy,wooden, RCC, east, west, north, south, low, high, very high, less,more,most, bad, good} = {et1, et2, et3, ef1, ef2, ef3, ef4, ep1, ep2, ep3, ed1, ed2, ed3, ec1, ec2}. The complement of swarm map fC ¬A : ¬V → P (U) is defined by fC ¬A(¬et1) = U − fA(et1) = U − {h2, h4, h9} = {h1, h3, h5, h6, h7, h8, h10}, i. e. fC ¬A(not muddy type houses) = {h1, h3, h5, h6, h7, h8, h10}, Similarly fC ¬A(¬et2) = fC ¬A(not wooden type houses) = {h2, h3, h4, h6, h9, h10}, fC ¬A(¬et3) = fC ¬A(not RCC type houses) = {h1, h2, h4, h5, h7, h8, h9}, fC ¬A(¬ef1) = fC ¬A(not east face houses) = {h1, h5, h9, h10}, fC ¬A(¬ef2) = fC ¬A(not west face houses) = U , fC ¬A(¬ef3) = fC ¬A(not north face houses) = {h2, h3, h4, h6, h7, h8}, fC ¬A(¬ef4) = fC ¬A(not south face houses) = U , fC ¬A(¬ep1) = fC ¬A(not low price houses) = {h1, h3, h5, h6, h7, h8, h10}, fC ¬A(¬ep2) = fC ¬A(not high price houses) = {h2, h3, h4, h6, h9, h10}, fC ¬A(¬ep3) = fC ¬A(not very high price houses) = {h1, h2, h4, h5, h7, h8, h9}, fC ¬A(¬ed1) = fC ¬A(not less durable houses) = {h1, h2, h3, h5, h6, h7, h8, h10}, fC ¬A(¬ed2) = fC ¬A(not more durable houses) = {h1, h3, h4, h6, h9}, fC ¬A(¬ed3) = fC ¬A(not most durable houses) = {h2, h4, h5, h7, h8, h9, h10}, fC ¬A(¬ec1) = fC ¬A(not bad condition houses) = {h1, h3, h6, h9, h10}, fC ¬A(¬ec2) = fC ¬A(not good condition houses) = {h2, h4, h5, h7, h8}, Definition 5.9: Null swarm set: A swarm set (fA, V ) is said to be Null swarm set denoted by ϕ, if ∀ ea ∈ VA, fA(ea) = ϕ (null-set). Example 5.10: Let U = {u1, u2, u3, u4, u5} be the set of objects and A = {P,Q,R} be the set attributes. The associate parameters sets of an attribute P is VP = {ep1, ep2, ep3}, Q is VQ = {eq1, eq2} and R is VR = {er1, er2, er3}. The collection of all associate parameter set is V = VP ∪ VQ ∪ VR = {ep1, ep2, ep3, eq1, eq2, er1, er2, er3}. The swarm map fA : V → P (U) is defined by fA(ep1) = {ϕ}, fA(ep2) = {ϕ}, fA(ep3) = {ϕ}, fA(eq1) = {ϕ}, fA(eq2) = {ϕ}, fA(er1) = {ϕ}, fA(er2) = {ϕ}, fA(er3) = {ϕ}. Then the swarm set is (fA, V ) = {(u1, {ϕ}), (u2, {ϕ}), (u3, {ϕ}), (u4, {ϕ}), (u5, {ϕ}). Therefore (fA, V ) is NULL swarm set. 6. Union and Intersecton of two swarm sets In this section, give definitions of union and intersecton of two swarm sets with examples. Let (fA, V ) and (fB , V ) are two swarm sets over common object set U and Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) https://internationalpubls.com 96 A, B are sets of attributes. The associate parameters sets are VA = ⋃ a∈A Va and VB = ⋃ b∈B Vb respectively. The collection of all associate parameter set is V = (∪a∈AVa) ⋃ (∪b∈BVb). Definition 6.1 : The union of (fA, V ) and (fB , V ) denoted by (fA, V )∪(fB , V ) and is defined by (fA, V ) ∪ (fB , V ) = (fA∪B , V ). The swarm map fA∪B : V → P (U) is defined by fA∪B(eα) =  fA(eα), if eα ∈ A−B fB(eα), if eα ∈ B −A fA(eα) ∪ fB(eα), if eα ∈ A ∩B (3) Example 6.2: Consider U = {h1, h2, h3, h4, h5, h6} is set of objects and A = {Type, Price,Durable} = {T, P,D}, B = {Type,Durable, Condition} = {T,D,C} are sets of attributes. The associate parameter set of an attribute A of Type is VT = {muddy, wooden} = {et1, et2}, Price is VP = {low, high} = {ep1, ep2} and Condi- tion is VC = {bad, good} = {ec1, ec2}. Also The associate parameter set of an attribute B of Type is VT = {muddy,wooden} = {et1, et2}, Durable is VD = {less,more} = {ed1, ed2} and Condition is VC = {bad, good} = {ec1, ec2}. Therefore the collection of all associated parameters set is V = {VT ∪ VP ∪ VD}∪{VT∪VD∪VC} = VT∪VP∪VD∪VC = {muddy,wooden, low, high, less,more, bad, good} = {et1, et2, ep1, ep2, ed1, ed2, ec1, ec2}. The swarm map fA : V → P (U) is defined by fA(et1) = {h2, h3}, fA(et2) = {h1, h4, h5, h6}, fA(ep1) = {h1, h3, h5}, fA(ep2) = {h2, h4, h6}. fA(ed1) = {h2, h4, h6}, fA(ed2) = {h1, h3, h5}. Then the swarm set is (fA, V ) = {(h1, {et2, ep1, ed2}), (h2, {et1, ep2, ed1}), (h3, {et1, ep1, ed2}), (h4, {et2, ep2, ed1}), (h5, {et2, ep1, ed2}), (h6, {et2, ep2, ed1})}. The swarm map fB : V → P (U) is defined by fB(et1) = {h1, h2, h3, h4}, fB(et2) = {h5, h6}, fB(ed1) = {h1, h5, h6}, fB(ed2) = {h2, h3, h4}, fB(ec1) = {h1, h4, h6}, fB(ec2) = {h2, h3, h5}. The swarm set is (fB , V ) = {(h1, {et1, ed1, ec1}), (h2, {et1, ed2, ec2}), (h3, {et1, ed2, ec2}), (h4, {et1, ed2, ec1}), (h5, {et2, ed1, ec2}), (h6, {et2, ed1, ec1})}. Here (fA, V ) and (fB , V ) are two swarm sets over a common object set U and A, B are sets of attributes. The collection of all associated parameters set V = {et1, et2, ep1, ep2, ed1, ed2, ec1, ec2}. If A − B = {T, P, D} − {T, D, C} = P , then the swarm map fA∪B : V → P (U) defined by [fA∪B(eα)] = fA(eα), ∀ eα ∈ A − B, such that fA∪B(ep1) = fA(ep1) = {h1, h3, h5}, fA∪B(ep2) = fA(ep2) = {h2, h4, h6} If B − A = {T, D, C} − {T, P, D} = C, then the swarm map fA∪B : V → P (U) defined by [fA∪B(eα)] = fB(eα), ∀ eα ∈ B − A, such that fA∪B(ec1) = fB(ec1) = {h1, h4, h6}, fA∪B(ec2) = fB(ec2) = {h2, h3, h5} If A ∩ B = {T, P, D} ∩ {T, D, C} = {T, D}, then the swarm map fA∪B : V → P (U) defined by [fA∪B(eα)] = fA(eα) ∪ fB(eα) , ∀ eα ∈ A ∩ B. fA∪B(et1) = fA(et1) ∪ fB(et1) = {h2, h3} ∪ {h1, h2, h3, h4} = {h1, h2, h3, h4} Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) https://internationalpubls.com 97 fA∪B(et2) = fA(et2) ∪ fB(et2) = {h1, h4, h5, h6} ∪ {h5, h6} = {h1, h4, h5, h6} fA∪B(ed1) = fA(ed1)∪fB(ed1) = {h2, h4, h6}∪{h1, h5, h6} = {h1, h2, h4, h5, h6} fA∪B(ed2) = fA(ed2)∪fB(ed2) = {h1, h3, h5}∪{h2, h3, h4} = {h1, h2, h3, h4, h5} Definition 6.3: The intersection of (fA, V ) and (fB , V ) denoted by (fA, V ) ∩ (fB , V ) and is defined by (fA, V ) ∩ (fB , V ) = (fA∩B , V ). The swarm map fA∩B : V → P (U) is defined by fA∩B(eα) =  fA(eα), if eα ∈ A−B fB(eα), if eα ∈ B −A fA(eα) ∩ fB(eα), if eα ∈ A ∩B (4) Example 6.4: By Exmple 6.2, (fA, V ) and (fB , V ) are two swarm sets over common object set U and A, B are sets of attributes. The collection of all asso- ciate parameters set is V = {et1, et2, ep1, ep2, ed1, ed2, ec1, ec2} If A−B = {T, P,D} − {T,D,C} = P , then the swarm map fA∩B : V → P (U) is defined by fA∩B(eα) = fA(eα), ∀ eα ∈ A−B, such that fA∩B(ep1) = fA(ep1) = {h1, h3, h5}, fA∩B(ep2) = fA(ep2) = {h2, h4, h6} If B −A = {T,D,C} − {T, P,D} = C, then the swarm map fA∩B : V → P (U) defined by [fA∩B(eα)] = fB(eα), ∀ eα ∈ B −A, such that fA∩B(ec1) = fB(ec1) = {h1, h4, h6}, fA∩B(ec2) = fB(ec2) = {h2, h3, h5} If A ∩B = {T, P,D} ∩ {T,D,C} = {T,D}, then the swarm map fA∩B : V → P (U) defined by [fA∩B(eα)] = fA(eα) ∩ fB(eα) , ∀ eα ∈ A ∩B [fA∩B(et1)] = [fA(et1)] ∩ [fB(et1)] = [{h2, h3}] ∩ [{h1, h2, h3, h4}] = [{h2, h3}] fA∩B(et2) = fA(et2) ∩ fB(et2) = {h1, h4, h5, h6} ∩ {h5, h6} = {h5, h6} fA∩B(ed1) = fA(ed1) ∩ fB(ed1) = {h2, h4, h6} ∩ {h1, h5, h6} = {h6} fA∩B(ed2) = fA(ed2) ∩ fB(ed2) = {h1, h3, h5} ∩ {h2, h3, h4} = {h3} Proposition 6.5: If (fA, V ) is a swarm set in the objects set U , A is set of attributes and the associate parameters set VA = ⋃ a∈A. The collection of all associate parameter set is V = (∪a∈AVa), then i) (fA, V ) ∪ (fA, V ) = (fA, V ) ii) (fA, V ) ∩ (fA, V ) = (fA, V ) Proposition 6.6: If (fA, V ) , (fB , V ) , (fC , V ) are three swarm sets over com- mon objects set U and A, B, C are sets of attributes. The associate parameters sets are VA = ⋃ a∈A Va, VB = ⋃ b∈B Vb and VC = ⋃ c∈C Vc. The collection of all associate parameter set is V = (∪a∈AVa) ⋃ (∪b∈BVb) ⋃ (∪c∈CVc), then i) [(fA, V ) ∪ (fB , V )] ∪ (fC , V ) = (fA, V ) ∪ [(fB , V ) ∪ (fC , V )] ii) [(fA, V ) ∩ (fB , V )] ∩ (fC , V ) = (fA, V ) ∩ [(fB , V ) ∩ (fC , V )] 7. Application of Swarm Set In this section, we present an application on real estate problem to illustrate the concept is discussed. The problem we consider is as below. In a town there are fifteen houses are in sale, meanwhile six customers came to purchase the house of their own requirements. Now by using swarm set it is possible for customers to select their own requirements for the house. Here three customers are officers, two customers of business persons and one customer is retired person. The customers requirements are as follows: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) https://internationalpubls.com 98 Customers X1, X2 and X3 are officers, their requirements are near by office, near by school, RCC type, medium dimension, east face, good location, good road and good water facility etc. Customers X4 and X5 are business persons, their requirements are near by market, near bus stand, RCC type, well dimension, north face, good location, good road and good water facility etc. Customer X6 is retired person, his requirements are outside the city, small dimension, east face, less price, RCC type, good location, good road and good water facility etc. Consider U = {h1, h2, h3, h4, h5, h6, h7, h8, h9, h10, h11, h12, h13, h14, h15} is object set of fifteen houses and A = {Type,Dimension, Face, Condition, Location, Price, School, Office,Road,Water,Bus stand,Market, City} = {T,D, F,Co, L, P, S,O,R,W,B,M,Ci} is set of attributes. The associate parameter set of an attribute, Type is VT = {wooden, RCC} = {et1, et2}, et1 stands for wooden type and et2 stands for RCC type. Similarly, VD = {20× 30, 30× 40, 40× 60} = {ed1, ed2, ed3}, ed1 stands for 20Ö30 dimen- sion, ed2 stands for 30Ö40 dimension and ed3 stands for 40Ö60 dimension. VF = {east, north} = {ef1, ef2}, ef1 stands for east face and ef2 stands for north face. VCo = {bad, good} = {eco1, eco2}, eco1 stands for bad condition and eco2 stands for good condition. VL = {bad, good} = {el1, el2}, el1 stands for bad location and el2 stands for good location. VP = {25 lakh , between 25 to 55 lakh ,more than 55 lakh} = {ep1, ep2, ep3}, ep1 stands for less price i.e less than 25 lakh, ep2 stands for costly price i.e be- tween 25 to 55 lakh and ep3 stands for every costly price i.e more than 55 lakh. VS = {near, long, very long} = {es1, es2, es3}, es1 stands for near by school i.e within 4 kms, es2 stands for long distance to school i.e between 4 to 10 kms and es3 stands for very long distance i.e above 10 kms to schools. VO = {near, long, very long} = {eo1, eo2, eo3}, eo1 stands for near by office i.e within 4 kms, eo2 stands for long distance to office i.e between 4 to 10 kms and eo3 stands for very long distance i.e above 10 kms to offices. VR = {bad, good, very good} = {er1, er2, er3}, er1 stands for bad road, er2 stands good road and er3 stands for very good road. VW = {bad, good, very good} = {ew1, ew2, ew3}, ew1 stands for bad water fa- cility, ew2 stands good water facility and ew3 stands for very good for water facility. VB = {near, long, very long} = {eb1, eb2, eb3}, eb1 stands for near to bus stand i.e within 4 kms, eb2 stands for long distance to bus stand i.e between 4 to 10 kms, eb3 stands for very long distance i.e above 10 kms to bus stand. VM = {near, long, very long} = {em1, em2, em3}, em1 stands for near to mar- ket i.e within 4 kms, em2 stands for long distance to market i.e between 4 to 10 kms, em3 stands for very long distance i.e above 10 kms to market. VCi = {inside, outside} = {eci1, eci2}, eci1 stands for inside the city and eci2 stands for outside the city. Therefore the collection of all associated parameters set V = VT ∪VD ∪VF ∪ Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) https://internationalpubls.com 99 VCo∪VL∪VP ∪VS ∪VO∪VR∪VW ∪VB ∪VM ∪VCi = {et1, et2, ed1, ed2, ed3, ef1, ef2, eco1, eco2, el1, el2, ep1, ep2, ep3, es1, es2, es3, eo1, eo2, eo3, er1, er2, er3, ew1, ew2, ew3, eb1, eb2, eb3, em1, em2, em3, eci1, eci2}. The swarm map fA : V → P (U) is defined by fA(wooden) = fA(et1) = {h1, h10, h15}, fA(RCC) = fA(et2) = {h2, h3, h4, h5, h6, h7, h8, h9, h11, h12, h13, h14}, fA(20 × 30) = fA(ed1) = {h1, h10, h15}, fA(30 × 40) = fA(ed2) = {h3, h4, h5, h7, h8, h9, h12, h13}, fA(40 × 60) = fA(ed3) = {h2, h6, h11, h14}, fA((east) = fA(ef1) = {h3, h4, h6, h7, h10, h12, h13, h14}, fA(north) = fA(ef2) = {h1, h2, h5, h8, h9, h11, h15}, fA(bad) = fA(eco1) = {h10, h15}, fA(good) = fA(eco2) = {h1, h2, h3, h4, h5, h6, h7, h8, h9, h11, h12, h13, h14}, fA(bad) = fA(el1) = {h1, h10}, fA(good) = fA(el2) = {h2, h3, h4, h5, h6, h7, h8, h9, h11, h12, h13, h14, h15}, fA(less) = fA(ep1) = {h1, h10, h15}, fA(coslty) = fA(ep2) = {h3, h4, h5, h7, h8, h9, h12, h13}, fA(very coslty) = fA(ep3) = {h2, h6, h11, h14}, fA(near) = fA(es1) = {h3, h5, h7, h8, h9, h12}, fA(long) = fA(es2) = {h2, h6, h11, h14}, fA(very long) = fA(es3) = {h1, h4, h10, h13, h15}, fA(near) = fA(eo1) = {h3, h5, h7, h8, h9, h12}, fA(long) = fA(eo2) = {h2, h6, h11, h14}, fA(very long) = fA(eo3) = {h1, h4, h10, h13, h15}, fA(bad) = fA(er1) = {h1, h10, h13, h15}, fA(good) = fA(er2) = {h4, h5, h8, h9, h11}, fA(very good) = fA(er3) = {h2, h3, h6, h7, h12, h14}, fA(bad) = fA(ew1) = {h1, h10, h13, h15}, fA(good) = fA(ew2) = {h4, h5, h11}, fA(very good) = fA(ew3) = {h2, h3, h6, h7, h8, h9, h12, h14}, fA(near) = fA(eb1) = {h2, h6, h11, h14}, fA(long) = fA(eb2) = {h3, h7, h9, h12, h15}, fA(very long) = fA(eb3) = {h1, h4, h5, h8, h10, h13}, fA(near) = fA(em1) = {h2, h6, h11, h14}, fA(long) = fA(em2) = {h3, h5, h7, h8, h9, h12, h15}, fA(very long) = fA(em3) = {h1, h4, h10, h13}, fA(inside) = fA(eci1) = {h2, h3, h5, h6, h7, h8, h9, h11, h12, h14}, fA(outside) = fA(eci2) = {h1, h4, h10, h13, h15}, Then the swarm set is (fA, V ) = {(h1 , { wooden type, 20 × 30 dimension, north face, good condition, bad location, less price, very long distance to school, very long distance to office, bad road, bad water facility, very long distance to bus stand, very long distance to market, outside the city }), (h2, { RCC type, 40 × 60 dimension, north face, good condition, good location, every costly price, long distance to school, long distance to office, very good road, very good water facility, near by bus stand, near by market, inside the city }), Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) https://internationalpubls.com 100 (h3, { RCC type, 30 × 40 dimension, east face, good condition, good location, costly price, near by school, near by office, very good road, very good water facility, long distance to bus stand, long distance to market, inside the city }), (h4, { RCC type, 30 × 40 dimension, east face, good condition, good location, costly price, very long distance to school, very long distance to office, good road, good water facility, very long distance to bus stand, very distance long to the market, outside the city }), (h5, { RCC type, 30 × 40 dimension, north face, good condition, good location, costly price, near by school, near by office, very good road, very good water facility, very long distance to bus stand, long distance to market, inside the city }), (h6, { RCC type, 40 × 60 dimension, east face, good condition, good location, every costly price, long distance to school, long distance to office, very good road, very good water facility, near by bus stand, near by market, inside the city }), (h7, { RCC type, 30 × 40 dimension, east face, good condition, good location, costly price, near by school, near by office, very good road, very good water facility, long distance to bus stand, long distance to market, inside the city }), (h8, { RCC type, 30 × 40 dimension, north face, good condition, good location, costly price, near by school, near by office, good road, very good water facility, very long distance to bus stand, long distance to market, inside the city }), (h9, { RCC type, 30 × 40 dimension, north face, good condition, good location, costly price, near by school, near by office, good road, very good water facility, long distance to bus stand, long distance to market, inside the city }), (h10, { wooden type, 20 × 30 dimension, east face, bad condition, bad location, less price, very long distance to school, very long distance to office, bad road, bad water facility, very long distance to bus stand, very distance long to market, outside the city }), (h11, { RCC type, 40 × 60 dimension, north face, good condition, good location, every costly price, long distance to school, long distance to office, good road, good water facility, near by bus stand, near by market, inside the city }), (h12, { RCC type, 30 × 40 dimension, east face, good condition, good location, costly price, near by school, near by office, very good road, very good water facility, long distance to bus stand, long distance to market, inside the city }), (h13, { RCC type, 30 × 40 dimension, east face, good condition, good location, costly price, very long distance to school, very long distance to office, bad road, bad water facility, very long distance to bus stand, very long distance to market, outside the city }), (h14, { RCC type, 40 × 60 dimension, east face, good condition, good location, every costly price, long distance to school, long distance to office, very good road, very good water facility, near by bus stand, near by market, inside the city }), (h15, { wooden type, 20 × 30 dimension, north face, bad condition, good loca- tion, less price, very long distance to school, very long distance to office, bad road, bad water facility, long distance to bus stand, long distance to market, outside the city }). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) https://internationalpubls.com 101 Tabular representation of swarm set is in Table 2. Table 2: Ob. Attributes U T D F Co L P S O R W B M Ci h1 et1 ed1 ef2 eco2 el1 ep1 es3 eo3 er1 ew1 eb3 em3 eci2 h2 et2 ed3 ef2 eco2 el2 ep3 es2 eo2 er3 ew3 eb1 em1 eci1 h3 et2 ed2 ef1 eco2 el2 ep2 es1 eo1 er3 ew3 eb2 em2 eci1 h4 et2 ed2 ef1 eco2 el2 ep2 es3 eo3 er2 ew2 eb3 em3 eci2 h5 et2 ed2 ef2 eco2 el2 ep2 es1 eo1 er2 ew2 eb3 em2 eci1 h6 et2 ed3 ef1 eco2 el2 ep3 es2 eo2 er3 ew3 eb1 em1 eci1 h7 et2 ed2 ef1 eco2 el2 ep2 es1 eo1 er3 ew3 eb2 em2 eci1 h8 et2 ed2 ef2 eco2 el2 ep2 es1 eo1 er2 ew3 eb3 em2 eci1 h9 et2 ed2 ef2 eco2 el2 ep2 es1 eo1 er2 ew3 eb2 em2 eci1 h10 et1 ed1 ef1 eco1 el1 ep1 es3 eo3 er1 ew1 eb3 em3 eci2 h11 et2 ed3 ef2 eco2 el2 ep3 es2 eo2 er2 ew2 eb1 em1 eci1 h12 et2 ed2 ef1 eco2 el2 ep2 es1 eo1 er3 ew3 eb2 em2 eci1 h13 et2 ed2 ef1 eco2 el2 ep2 es3 eo3 er1 ew1 eb3 em3 eci2 h14 et2 ed3 ef1 eco2 el2 ep3 es2 eo2 er3 ew3 eb1 em1 eci1 h15 et1 ed1 ef2 eco1 el2 ep1 es3 eo3 er1 ew1 eb2 em2 eci2 Find the requirements of customers by using the formula Requirement = ⋂ {fA(parameter)}customer requirement = ⋂ α∈A {fA(α), α ∈ requirement parameter } Officers requirement = ⋂ {fA(parameters)}requirements = ⋂ α∈A {fA(α), α ∈ requirements parameters} = ∩ (near by office) ∩ (near by school) ∩ (medium dimension) ∩ (granite type) ∩ (east facing) ∩ (wide road) ∩ (sufficient water facility) = ∩fA(eo1) ∩ fA(es1) ∩ fA(ed2) ∩ fA(et2) ∩ fA(ef1) ∩ fA(er3) ∩ fA(ew3) = ∩{h3, h5, h7, h8, h9, h12} ∩ {h3, h5, h7, h8, h9, h12} ∩ {h3, h5, h7, h8, h9, h12} ∩{h2, h3, h4, h5, h6, h7, h8, h9, h11, h12, h13, h14}∩{h3, h4, h5, h7, h8, h9, h12, h13} ∩{h2, h3, h4, h5, h6, h7, h8, h9, h11, h12, h13, h14}∩{h2, h3, h4, h5, h6, h7, h8, h9, h11, h12, h13, h14} = {h3, h5, h7, h8, h9, h12} Officers customers can select their own house which meets their require- ments by using swarm set, because each house information is available. Here h3, h5, h7, h8, h9 and h12 houses are suitable for officers X1, X2 and X3. So in this concept the customers X1, X2 and X3 can select any one of the house in the list {h3, h5, h7, h8, h9, h12}. Business Pepole requirement = ⋂ {fA(parameters)}requirements = ⋂ α A {fA(α), α ∈ requirements parameters} = ∩ (near by mark ∈ et) ∩ (near by bus stand) ∩ (wide dimension) ∩ (granite type) ∩ (north facing) ∩ (wide road) ∩ (sufficient water facility) = ∩fA(em1) ∩ fA(eb1) ∩ fA(ed3) ∩ fA(et2) ∩ fA(ef2) ∩ fA(er3) ∩ fA(ew3) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) https://internationalpubls.com 102 = ∩{h2, h6, h11, h14}∩{h2, h6, h11, h14}∩{h2, h6, h11, h14}∩{h2, h3, h4, h5, h6, h7, h8, h9, h11, h12, h13, h14}∩{h1, h2, h6, h10, h11, h14, h15}∩{h2, h3, h4, h5, h6, h7, h8, h9, h11, h12, h13, h14} ∩ {h2, h3, h4, h5, h6, h7, h8, h9, h11, h12, h13, h14} = {h2, h6, h11, h14} Business Pepole can select their own house which meets their requirements by using swarm set. Here h2, h6, h11 and h14 houses are suitable for business customers X4 and X5. So in this concept the customers X4 and X5 can select any one of the house in the list {h2, h6, h11, h14}. Retired person requirement = ⋂ {fA(parameters)}requirements = ⋂ α∈A {fA(α), α ∈ requirements parameters} = ∩ (outside city) ∩ (small dimension) ∩ (east facing) ∩ (less price) ∩ (wide road) ∩ (sufficient water facility) = ∩fA(eci2) ∩ fA(ed1) ∩ fA(ef1) ∩ fA(ep1) ∩ fA(eco2) ∩ fA(er3) ∩ fA(ew3) = ∩{h1, h4, h10, h13} ∩ {h1, h4, h10, h13, h15} ∩ {h3, h4, h5, h7, h8, h9, h12, h13} ∩{h1, h4, h10, h13, h15}∩{h2, h3, h4, h5, h6, h7, h8, h9, h11, h12, h13, h14}∩{h2, h3, h4, h5, h6, h7, h8, h9, h11, h12, h13, h14} ∩ {h2, h3, h4, h5, h6, h7, h8, h9, h11, h12, h13, h14} = {h4, h13} Retired person can select own house which meets his requirements by using swarm set. Here h4 and h13 houses are suitable for retired person X6. So in this concept the customer X6 can select any one of the house in the list {h4, h13}. 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