Ratio Mathematica Volume 42, 2022 Some characteristics of picture fuzzy subgroups via cut set of picture fuzzy set Taiwo Olubunmi Sangodapo* Babatunde Oluwaseun Onasanya‡ Abstract Given any picture fuzzy set Q of a universe Y , the set Cr,s,t(Q) called the (r, s, t)-cut set of Q was studied. In this paper, some characteris- tics of picture fuzzy subgroup of a group are obtained via the cut sets of picture fuzzy set. Keywords: fuzzy set, picture fuzzy set, picture fuzzy subgroup, cut set 2020 AMS subject classifications: 20N25, 08A72, 03E72.1 *Department of Mathematics, Faculty of Science (University of Ibadan, Oyo State, Nigeria); to.ewuola@ui.edu.ng, toewuola77@gmail.com. †Corresponding Author; Department of Mathematics, Faculty of Science (University of Ibadan, Oyo State, Nigeria); babtu2001@yahoo.com, bo.onasanya@ui.edu.ng. 1 10.23755/rm.v39i0.849. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 341 † Received on May 24th, 2022. Accepted on June 29th, 2022. Published on June 30th, 2022. doi: 1 Introduction Zadeh [18] introduced the notion of fuzzy sets (FSs). This theory has been generalised by many researchers. Atanassov [1] initiated the concept of intuition- istic fuzzy sets (IFSs). For more on intuitionistic fuzzy set, also see [2, 3, 4]. The application of these fuzzy sets have taken different directions such as optimization and decision-making [15] control theory [13] and many others. One particular ap- plication is the work of Cuong and Kreinovich [5] who put forward the notion of picture fuzzy sets (PFSs) as a generalisation of fuzzy sets and intuitionistic fuzzy sets. Picture fuzzy set has been extensively studied and applied (see [6, 7, 8, 9, 10, 11, 12, 14, 17] for details). Rosenfeld [16] generalised fuzzy sets to fuzzy groups (FGs). The idea of cut set of picture fuzzy sets was initiated by Dutta and Ganju [12] and they obtained some of its properties. Dogra and Pal [11] corrected the definition of cut set that was given by Dutta and Ganju [12] and they initiated the notion of picture fuzzy subgroups (PFSGs) of a group. In this paper, motivated by the work of Dogra and Pal [11], investigation of the characteristics of picture fuzzy subgroups of a group via the cut sets of picture fuzzy sets was done. The organisation of the paper is as follows: Section 2 gives the basic definitions and preliminary ideas of PFSs; Section 3 investigates some properties of picture fuzzy subgroups of a group via cut sets of picture fuzzy sets. 2 Preliminaries Definition 2.1. [18] Let Y be a nonempty set. A FS Q of Y is an object of the form Q = {⟨y, σQ(y)⟩|y ∈ Y } with a membership function σQ : Y −→ [0, 1] where the function σQ(y) denotes the degree of membership of y ∈ Q. Definition 2.2. [1] Let a nonempty set Y be fixed. An IFS Q of Y is an object of the form Q = {⟨y, σQ(y), τQ(y)⟩|y ∈ Y } where the functions σQ : Y → [0, 1] and τQ : Y → [0, 1] are called the membership and non-membership degrees of y ∈ Q, respectively, and for every y ∈ Y , 0 ≤ σQ(y) + τQ(y) ≤ 1. 342 Picture fuzzy subgroups via cut set of picture fuzzy set Definition 2.3. [5] A picture fuzzy set Q of Y is defined as Q = {(y, σQ(y), τQ(y), γQ(y))|y ∈ Y }, where the functions σQ : Y → [0, 1], τQ : Y → [0, 1] and γQ : Y → [0, 1] are called the positive, neutral and negative membership degrees of y ∈ Q, re- spectively, and σQ, τQ, γQ satisfy 0 ≤ σQ(y) + τQ(y) + γQ(y) ≤ 1, ∀y ∈ Y. For each y ∈ Y , SQ(y) = 1 − (σQ(y) + τQ(y) + γQ(y)) is called the refusal membership degree of y ∈ Q. Definition 2.4. [5] Let Q and R be two PFSs. Then, the inclusion, equality, union, intersection and complement are defined as follow: (i) Q ⊆ R if and only if for all y ∈ Y , σQ(y) ≤ σR(y), τQ(y) ≤ τR(y) and γQ(y) ≥ γR(y). (ii) Q = R if and only if Q ⊆ R and R ⊆ Q. (iii) Q ∪ R = {(y, σQ(y) ∨ σR(y), τQ(y) ∨ τR(y)), γQ(y) ∧ γR(y))|y ∈ Y }. (iv) Q ∩ R = {(y, σQ(y) ∧ σR(y), τQ(y) ∧ τR(y)), γQ(y) ∨ γR(y))|y ∈ Y }. Definition 2.5. [11] Let (G, ∗) be a crisp group and Q = {(y, σQ(y), τQ(y), ηQ(y)) | y ∈ G} be a PFS in G. Then, Q is called picture fuzzy subgroup of G (PFSG) if (i) σQ(a∗b) ≥ σQ(a)∧σQ(b), τQ(a∗b) ≥ τQ(a)∧τQ(b), ηQ(a∗b) ≤ ηQ(a)∨ηQ(b) (ii) σQ(a−1) ≥ σQ(a), τQ(a−1) ≥ τQ(a), ηQ(a−1) ≤ ηQ(a) for all a, b ∈ G. Notice that a−1 is the inverse of a ∈ G, or equivalently, Q is a PFSG of G if and only if σQ(a ∗ b−1) ≥ σQ(a) ∧ σQ(b), τQ(a ∗ b−1) ≥ τQ(a) ∧ τQ(b), ηQ(a ∗ b−1) ≤ ηQ(a) ∨ ηQ(b). Definition 2.6. [11] Let (G, ∗) be a crisp group and Q = (σQ, τQ, ηQ) be a PFSG of G. Then, for a ∈ G the picture fuzzy left coset of Q ∈ G is the PFS aQ = (σaQ, τaQ, ηaQ) defined by σaQ(u) = σQ(a −1 ∗ u), τaQ(u) = τQ(a−1 ∗ u) and ηaQ(u) = ηQ(a−1 ∗ u) for all u ∈ G. 343 Definition 2.7. [11] Let (G, ∗) be a crisp group and Q = (σQ, τQ, ηQ) be a PFSG of G. Then, for a ∈ G the picture fuzzy right coset of Q ∈ G is the PFS Qa = (σQa, τQa, ηQa) defined by σQa(u) = σQ(u ∗ a−1), τQa(y) = τQ(u ∗ a−1) and ηQa(u) = ηQ(u ∗ a−1) for all u ∈ G. Definition 2.8. [11] Let (G, ∗) be a crisp group and Q = (σQ, τQ, ηQ) be a PFSG of G. Then, Q is called a picture fuzzy normal subgroup (PFNSG) of G if σQa(y) = σaQ(y), τQa(y) = τaQ(y), ηQa(y) = ηaQ(y) for all a, y ∈ G. Remark 2.1. Dogra and Pal established that PFSG of G is normal if and only if (i) σQ(u −1 ∗ a ∗ u) = σQ(a) (ii) τQ(u −1 ∗ a ∗ u) = τQ(a) (ii) ηQ(u −1 ∗ a ∗ u) = ηQ(a). For all a ∈ Q and u ∈ G. Cut set of picture fuzzy sets was introduced by Dutta and Ganju ? but the definition did not capture the cut set very well. Thus, Dogra and Pal ? corrected it in their paper. Definition 2.9. [11] Let Q = {(x, σQ, τQ, γQ)|a ∈ Y } be PFS over the universe Y . Then, (r, s, t)-cut set of Q is the crisp set in Q, denoted by Cr,s,t(Q) and is defined by Cr,s,t(Q) = {a ∈ Y |σQ(a) ≥ r, τQ(a) ≥ s, γQ(a) ≤ t} r, s, t ∈ [0, 1] with the condition 0 ≤ r + s + t ≤ 1. Theorem 2.1. [11] Let (G, ∗) be a crisp group and Q = (σQ, τQ, ηQ) be a PFSG of G. Then, Q is a PFSG if and only if Cr,s,t(Q) is a crisp subgroup of G. Theorem 2.2. [12] If Q and R are two PFSs of a universe Y , then the following results hold (i) Cr,s,t(Q) ⊆ Cu,v,w(Q) if r ≥ u, s ≤ v, t ≤ w. (ii) C1−s−t,s,t(Q) ⊆ Cr,s,t(Q) ⊆ Cr,1−r−t,t(Q). (iii) Q ⊆ R implies Cr,s,t(Q) ⊆ Cr,s,t(R). 344 Picture fuzzy subgroups via cut set of picture fuzzy set (iv) Cr,s,t(Q ∩ R) = Cr,s,t(Q) ∩ Cr,s,t(R). (v) Cr,s,t(Q ∪ R) ⊇ Cr,s,t(Q) ∪ Cr,s,t(R). (vi) Cr,s,t(∩Qi) = ∩Cr,s,t(Qi). (vii) C1,0,0(Q) = Y. 3 Main Results Remark 3.1. It is important to note that [12] misquoted the result of [5]. Hence, the results built on this foundation cannot be or at all correct. The counter exam- ple in Example (3.1) shows that the claims in Theorem 2.2 (i), (ii) and (vii) are wrong. The correct version of Theorem (2.2) is given in Theorem (3.1). Example 3.1. Let Y = {y1, y2, y3, y4}, Q = {(y1, 0, 0.1, 0.8), (y2, 0.2, 0.5, 0.3), (y3, 0.4, 0.2, 0.1), (y4, 0.5, 0.3, 0.2)}, and R = {(y1, 0.1, 0.4, 0.5), (y2, 0.3, 0.6, 0.1), (y3, 0.5, 0.3, 0), (y4, 0.5, 0.4, 0.1)} be PFSs, taking r = 0.1, s = 0.3, t = 0.5. Thus, the (0.1, 0.3, 0.5)-cut set of Q are C0.1,0.3,0.5(Q) = {y2, y4} and C0.1,0.3,0.5(R) = {y1, y2, y3, y4}. Hence, C1−s−t,s,t(Q) = C0.2,0.3,0.5(Q) = {y2, y4}, Cr,s,t(Q) = C0.1,0.3,0.5(Q) = {y2, y4}, C0.1,0.2,0.3(Q) = {y3} and Cr,1−r−t,t(Q) = C0.1,0.4,0.5(Q) = {y2}. Thus, C1−s−t,s,t(R) ⊆ Cr,s,t(R) ⊊ Cr,1−r−t,t(R), which means (i) and (ii) fail. 345 Note that the left side of the inclusion which holds obey the condition (i) of our Theorem 3.1 Furthermore, C1−s−t,s,t(Q) ⊆ Cr,s,t(Q) ⊊ Cr,1−r−t,t(Q), which means (i) and (ii) fail. Note that the left side of the inclusion which holds obey the condition (i) of our Theorem 3.1 In addition, C0.1,0.2,0.3(Q) = {y3} ⊊= {y2, y4} = C0.1,0.3,0.5(Q) which means (i) and (ii) fail. Also, C1,0,0(Q) = ∅ ≠ Y , which means (vii) fails. Theorem 3.1. Let Q and R be two PFSs of a universe Y . Then, the following assertions hold: (i) Cr,s,t(Q) ⊆ Cu,v,w(Q) if r ≥ u, s ≥ v, t ≤ w. (ii) C1−s−t,s,t(Q) ⊆ Cr,s,t(Q) ⊆ Cr,s,1−r−s(Q), (iii) C1−s−t,1−r−t,t(Q) ⊆ Cr,s,t(Q) ⊆ Cr,s,1−r−s(Q), (iv) Cr,1−r−t,t(Q) ⊆ Cr,s,t(Q) ⊆ Cr,s,1−r−s(Q), (v) C0,0,1(Q) = Y. Proof. (i) Let x ∈ Cr,s,t(Q). Using Definition 2.9, σQ(x) ≥ r ≥ u, τQ(x) ≥ s ≥ v, γQ(x) ≤ t ≤ w. Thus, x ∈ Cu,v,w(Q) and, as such, Cr,s,t(Q) ⊆ Cu,v,w(Q). (ii) Since r + s + t ≤ 1 implies that 1 − s − t ≥ r, s ≥ s and t ≤ t, and r ≥ r, s ≥ s and t ≤ 1 − r − s, by Theorem 3.1 (i), the result holds. (iii) Since r + s + t ≤ 1 implies that 1 − s − t ≥ r, 1 − r − t ≥ s and t ≤ t, and r ≥ r, s ≥ s and t ≤ 1 − s − r, by Theorem 3.1 (i), the result holds. (iv) Since r + s + t ≤ 1 implies that r ≥ r, 1 − r − t ≥ s and t ≤ t, and r ≥ r, s ≥ s and t ≤ 1 − s − r, by Theorem 3.1 (i), the result holds. (v) Since ∀ y ∈ Y, σ(y) ≥ 0, τ(y) ≥ 0, γ(y) ≤ 1, C0,0,1(Q) = Y , 346 Picture fuzzy subgroups via cut set of picture fuzzy set Proposition 3.1. If Q is PFSG of G, then Cr,s,t(Q) is a subgroup of G, where σQ(e) ≥ r, τQ(e) ≥ s, and ηQ(e) ≤ t and e is the identity element of G. Proof. Clearly Cr,s,t(Q) ̸= ∅ as e ∈ Cr,s,t(Q). Let a, b ∈ Cr,s,t(Q) be any two elements. Then, σQ(a) ≥ r, τQ(a) ≥ s, ηQ(a) ≤ t and σQ(b) ≥ r, τQ(b) ≥ s, ηQ(b) ≤ t σQ(a) ∧ σQ(b) ≥ r, τQ(a) ∧ τQ(b) ≥ s and ηQ(a) ∨ ηQ(b) ≤ t Since Q is a PFSG of G, σQ(a ∗ b−1) ≥ σQ(a) ∧ σQ(b) ≥ r, τQ(a ∗ b−1) ≥ τQ(a) ∧ τQ(b) ≥ s and ηQ(a ∗ b−1) ≤ ηQ(a) ∨ ηQ(b) ≤ t. Hence, a ∗ b−1 ∈ Cr,s,t(Q) and Cr,s,t(Q) is a subgroup of G. Proposition 3.2. If Q is PFNSG of G, then Cr,s,t(Q) is a normal subgroup of G, where σQ(e) ≥ r, τQ(e) ≥ s, and ηQ(e) ≤ t and e is the identity element of G. Proof. Let a ∈ Cr,s,t(Q) and u ∈ G be any element. Then, σQ(a) ≥ r, τQ(a) ≥ s, ηQ(a) ≤ t. Also, since Q is a PFNSG of G, σQ(u −1∗a∗u) = σQ(a) ≥ r, τQ(u−1∗a∗u) = τQ(a) ≥ s, and ηQ(u−1∗a∗u) = ηQ(a) ≤ t ∀ a ∈ Q and u ∈ G. Therefore, u−1 ∗ a ∗ u ∈ Cr,s,t(Q), so a ∗ u ∈ u ∗ Cr,s,t(Q) which implies Cr,s,t(Q)∗u ⊆ u∗Cr,s,t(Q). Also, u∗a∗u−1 ∈ Cr,s,t(Q), so u∗a ∈ Cr,s,t(Q) ∗ u which implies u ∗ Cr,s,t(Q) ⊆ Cr,s,t(Q) ∗ u. Hence, u ∗ Cr,s,t(Q) = Cr,s,t(Q) ∗ u. Thus, Cr,s,t(Q) is a normal subgroup of G. Proposition 3.3. Given two PFSGs Q and R of a group (G, ∗). Then, Q ∩ R is a PFSG of G. This proposition has been proved by Dogra and Pal [11] by using PFSG definition. But, in this paper, a rather simpler approach of cut set of PFS is used to prove it. Proof. By Theorem 2.1, Q ∩ R is a PFSG of G if and only if Cr,s,t(Q ∩ R) is a crisp subgroup of G. Since Cr,s,t(Q ∩ R) = Cr,s,t(Q) ∩ Cr,s,t(R) (Theorem 2.2, iv) and both Cr,s,t(Q) and Cr,s,t(R) are subgroups of G and intersection of two subgroups of a group is its subgroup, Cr,s,t(Q∩R) is a subgroup of G. Therefore, Q ∩ R is a PFSG of G. 347 Notice that the union of two PFSGs of G need not be a PFSG of G [11]. Proposition 3.4. Let Q be a PFSG of G and a be any fixed element of G. Then, (i.) a ∗ Cr,s,t(Q) = Cr,s,t(a ∗ Q). (ii.) Cr,s,t(Q) ∗ a = Cr,s,t(Q ∗ a), ∀ r, s, t [0, 1] with r + s + t ≤ 1. Proof. (i.) Cr,s,t(a ∗ Q) = {u ∈ G|σa∗Q(u) ≥ r, τa∗Q(u) ≥ s, ηa∗Q(u) ≤ t} with the condition 0 ≤ r + s + t ≤ 1. Also a ∗ Cr,s,t(Q) = a ∗ {b ∈ G|σQ(b) ≥ r, τQ(b) ≥ s, ηQ(b) ≤ t} = {a ∗ b ∈ G|σQ(b) ≥ r, τQ(b) ≥ s, ηQ(b) ≤ t} Set a ∗ b = u, so that b = a−1 ∗ u. Therefore, a ∗ Cr,s,t(Q) = { u ∈ G|σQ(a−1 ∗ u) ≥ r, τQ(a−1 ∗ u) ≥ s, ηQ(a−1 ∗ u) ≤ t } = {u ∈ G|σa∗Q(u) ≥ r, τa∗Q(u) ≥ s, ηa∗Q(u) ≤ t} = Cr,s,t(a ∗ Q) for all r, s, t [0, 1] with r + s + t ≤ 1. (ii.) Cr,s,t(Q ∗ a) = {u ∈ G|σQ∗a(u) ≥ r, τQ∗a(u) ≥ s, ηQ∗a(u) ≤ t} with the condition 0 ≤ r + s + t ≤ 1. Also Cr,s,t(Q ∗ a) = {b ∈ G|σQ(b) ≥ r, τQ(b) ≥ s, ηQ(b) ≤ t} ∗ a = {b ∗ a ∈ G|σQ(b) ≥ r, τQ(b) ≥ s, ηQ(b) ≤ t} 348 Picture fuzzy subgroups via cut set of picture fuzzy set Set b ∗ a = u, so that b = u ∗ a−1. Therefore, Cr,s,t(Q) ∗ a = { u ∈ G|σQ(u ∗ a−1) ≥ r, τQ(u ∗ a−1) ≥ s, ηQ(u ∗ a−1) ≤ t } = {u ∈ G|σQ∗a(u) ≥ r, τQ∗a(u) ≥ s, ηQ∗a(u) ≤ t} = Cr,s,t(Q ∗ a) for all r, s, t [0, 1] with r + s + t ≤ 1. Proposition 3.5. Let Q be a PFSG of G. Let a, b ∈ G such that σQ(a) ∧ σQ(b) = r, τQ(a) ∧ τQ(b) = s, ηQ(a) ∨ ηQ(b) = t. Then, (i.) a ∗ Q = b ∗ Q ⇔ a−1 ∗ b ∈ Cr,s,t(Q) (ii.) Q ∗ a = Q ∗ b ⇔ a ∗ b−1 ∈ Cr,s,t(Q) Proof. (i.) a ∗ Q = b ∗ Q ⇔ Cr,s,t(a ∗ Q) = Cr,s,t(b ∗ Q) ⇔ a ∗ Cr,s,t(Q) = b ∗ Cr,s,t(Q) [by Proposition 3.4] ⇔ a−1∗b ∈ Cr,s,t(Q) [since Cr,s,t(Q) is a subgroup of G and a, b ∈ Cr,s,t(Q)]. (ii.) 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