Starshaped (∈,∈ ∨  q )-fuzzy sets

1 Introduction

In [1], Brown introduced the concept of starshaped fuzzy sets, and in [2] Diamond defined another type of starshaped fuzzy sets and established some of the basic properties of this family of fuzzy sets. Brown’s starshaped fuzzy sets was called quasi-starshaped fuzzy sets, and provided basic properties of them in the paper [5].

Zhan et al. [7] applied rough set theory to hemirings. Also, Ma et al. introduced the concepts of (∈  _γ , ∈  _γ  ∨ q _δ )-fuzzy h-bi-(h-quasi-) ideals of hemirings. They provided some new characterization theorems of these kinds of fuzzy h-ideals.

In this paper, using the notion of fuzzy points, we consider a generalization of starshaped fuzzy sets and quasi-starshaped fuzzy sets. We introduce the concepts of starshaped (∈, ∈  ∨ q)-fuzzy sets and quasi-starshaped (∈, ∈  ∨ q)-fuzzy sets, and investigate related properties. We provide characterizations of starshaped (∈, ∈  ∨ q)-fuzzy sets and quasi-starshaped (∈, ∈  ∨ q)-fuzzy sets.

2 Preliminaries

A fuzzy set λ in a set X of the form λ ( y ) : = { t ∈ ( 0 , 1 ] if y = x , 0 if y ≠ x , is said to be a fuzzy point with support x and value t and is denoted by x _t  . For a fuzzy point x _t and a fuzzy set λ in a set X, we say that x _t  ∈ λ (resp. x _t qλ) if λ (x) ≥ t (resp. λ (x) + t > 1), and in this case, x _t is said to belong to (resp. be quasi-coincident with) a fuzzy set λ (see [4]).

Let ℝ ⁿ denote the n-dimensional Euclidean space. For x, y ∈ ℝ ⁿ , the line segment xy ¯ joining x and y is the set of all points of the form αx + βy where α ≥ 0, β ≥ 0 and α + β = 1. A set S ⊆ ℝ ⁿ is said to be starshaped related to a point x ∈ ℝ ⁿ if xy ¯ ⊆ S for each point y ∈ S. A set S ⊆ ℝ ⁿ is simply said to be starshaped if there exists a point x in ℝ ⁿ such that S is starshaped relative to it. Note that a star-shaped set is not necessarily convex in the ordinary sense.

A fuzzy set λ ∈ ℱ(ℝ ⁿ ) is called a starshaped fuzzy set relative to y ∈ ℝ ⁿ (see [5, 6]) if it satisfies: ( ∀ x ∈ ℝ n ) ( ∀ k ∈ [ 0 , 1 ] ) ( λ ( k ( x - y ) + y ) ≥ λ ( x ) ) .(2.1)

A fuzzy set λ ∈ ℱ (ℝⁿ) is called a quasi-starshaped fuzzy set relative to y ∈ ℝ ⁿ (see [1, 5]) if it satisfies: ( ∀ x ∈ ℝ n ) ( ∀ k ∈ [ 0 , 1 ] ) ( λ ( kx + ( 1 - k ) y ) ≥ min { λ ( x ) , λ ( y ) } ) .(2.2)

In what follows, let  ℱ (ℝⁿ) denote the class of fuzzy sets on ℝ ⁿ .

Definition 3.1. A fuzzy set λ ∈ ℱ (ℝⁿ) is called a starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ if x t ∈ λ ⇒ ( k ( x - y ) + y ) t ∈ ∨ q λ(3.1) for all x ∈ ℝ ⁿ , k ∈ [0, 1] and t ∈ (0, 1].

Example 3.2. The fuzzy set λ ∈ ℱ (ℝ) given by λ : ℝ → [ 0 , 1 ] , x ↦ { e x if x ∈ ( - ∞ , 0 ] , e - x if x ∈ ( 0 , ∞ ) is a starshaped (∈, ∈ ∨ q)-fuzzy set relative to y = 0.

Example 3.3.The fuzzy set λ ∈ ℱ (ℝ) given by λ : ℝ → [ 0 , 1 ] , x ↦ { 1.5 + x if x ∈ ( - 1.5 , - 0.5 ] , 0.5 - x if x ∈ ( - 0.5 , 0 ] , 0.5 + x if x ∈ ( 0 , 0.5 ] , 1.5 - x if x ∈ ( 0.5 , 1.5 ) , 0 otherwise , is a starshaped (∈, ∈ ∨ q)-fuzzy set relative to y = 0.

Obviously, every starshaped fuzzy set relative to y ∈ ℝ ⁿ is a starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ , but the converse is not true. In fact, the starshaped (∈, ∈ ∨ q)-fuzzy set λ relative to y = 0 in Example 3.3 is not a starshaped fuzzy set relative to y = 0 since if x ∈ (-0.5, 0), then kx > x for k ∈ (0, 1) and λ (kx) < λ (x). Or, if x ∈ (0, 0.5) then kx < x for k ∈ (0, 1) and λ (kx) < λ (x).

We provide a condition for a fuzzy set λ ∈ ℱ (ℝⁿ) to be a starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ .

Theorem 3.4. Given y ∈ ℝ ⁿ , if a fuzzy set λ ∈ ℱ (ℝ ⁿ ) satisfies the condition x t ∈ ∨ q λ ⇒ ( k ( x - y ) + y ) t ∈ ∨ q λ(3.2) for all x ∈ ℝ ⁿ , k ∈ [0, 1] and t ∈ (0, 1], then λ is a starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ .

Proof. Straightforward.□

We consider characterizations of a starshaped (∈, ∈ ∨ q)-fuzzy set.

Theorem 3.5. A fuzzy set λ ∈ ℱ (ℝⁿ) is a starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ if and only if ( ∀ x ∈ ℝ n ) ( ∀ k ∈ [ 0 , 1 ] ) ( λ ( k ( x - y ) + y ) ≥ min { λ ( x ) , 0.5 } )(3.3)

Proof. Assume that λ is a starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ . Let x ∈ ℝ ⁿ and k ∈ [0, 1], and suppose that λ (x) <0.5. If λ (k (x - y) + y) < λ (x), then λ (k (x - y) + y) < t < λ (x) for some t ∈ (0, 0.5) and so x _t  ∈ λ but (k (x - y) + y)  _t oiλ. Since λ (k (x - y) + y) + t < 1, we have (k (x - y) + y)  _t oqλ. Hence ( k ( x - y ) + y ) t ∈ ∨ q ¯ λ, a contradiction. Thus λ (k (x - y) + y) ≥ λ (x). Now if λ (x) ≥0.5, then x_0.5 ∈ λ and so (k (x - y) + y) _0.5∈ ∨ q λ. Thus λ (k (x - y) + y) ≥0.5. Otherwise, λ (k (x - y) + y) +0.5 < 1, a contradiction. Consequently, λ (k (x - y) + y) ≥ min {λ (x) , 0.5} for all x ∈ ℝ ⁿ and k ∈ [0, 1].

Conversely, assume that a fuzzy set λ ∈ ℱ (ℝⁿ) satisfies the condition (3.3). Let x ∈ ℝ ⁿ , k ∈ [0, 1] and t ∈ (0, 1] be such that x _t  ∈ λ. Then λ (x) ≥ t. Suppose that λ (k (x - y) + y) < t. If λ (x) <0.5, then λ (k (x - y) + y) ≥ min {λ (x) , 0.5} = λ (x) ≥ t, a contradiction. Hence λ (x) ≥0.5, and so λ ( k ( x - y ) + y ) + t > 2 λ ( k ( x - y ) + y ) ≥ 2 min { λ ( x ) , 0.5 } = 1 .

Thus (k (x - y) + y)  _t ∈ ∨ q λ. Therefore λ is a starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ .□

Theorem 3.6. A fuzzy set λ ∈ ℱ (ℝⁿ) is a starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ if and only if its nonempty t-level set U (λ ; t) is starshaped relative to y ∈ ℝ ⁿ for all t ∈ (0, 0.5].

Proof. Suppose that the nonempty t-level set U (λ ; t) is starshaped relative to y ∈ ℝ ⁿ for all t ∈ (0, 0.5]. For k ∈ [0, 1] and x ∈ ℝ ⁿ , let λ (x) = t _x . Then xy ¯ ⊆ U ( λ ; t x ), and so λ ( k ( x - y ) + y ) ≥ t x = λ ( x ) ≥ min { λ ( x ) , 0.5 } . It follows from Theorem 3.5 that λ is a starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ .

Conversely, let a fuzzy set λ ∈ ℱ (ℝⁿ) be a starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ , and assume that U (λ ; t)≠ ∅ for all t ∈ (0, 0.5]. Let x ∈ U (λ ; t). Then λ (x) ≥ t, and so λ (k (x - y) + y) ≥ min {λ (x) , 0.5} ≥ min {t, 0.5} = t by Theorem 3.5. Hence xy ¯ ⊆ U ( λ ; t ) for t ∈ (0, 0.5]. Therefore U (λ ; t) is starshaped relative to y ∈ ℝ ⁿ for all t ∈ (0, 0.5].□

Theorem 3.7. For a starshaped fuzzy set λ ∈ ℱ (ℝⁿ), the nonempty t-level set U (λ ; t) is a starshaped subset of ℝ ⁿ relative to y ∈ ℝ ⁿ for all t ∈ (0.5, 1] if and only if λ satisfies the following condition. λ ( x ) ≤ max { λ ( k ( x - y ) + y ) , 0.5 }(3.4) for all x ∈ ℝ ⁿ and k ∈ [0, 1].

Proof. Assume that the nonempty t-level set U (λ ; t) is starshaped relative to y ∈ ℝ ⁿ for all t ∈ (0.5, 1]. If the condition 3.4 is false, then there exists a ∈ ℝ ⁿ such that λ (a) > max {λ (k (a - y) + y) , 0.5}. Hence t _a  : = λ (a) ∈ (0.5, 1] and a ∈ U (λ ; t _a ). But λ (k (a - y) + y) < t _a implies that ay ¯ ⊈ U ( λ ; t a ), that is, U (λ ; t _a ) is not a starshaped subset of ℝ ⁿ relative to a ∈ ℝ ⁿ . This is a contradiction, and so the condition 3.4 is valid.

Conversely, suppose that λ satisfies the condition 3.4. For any k ∈ [0, 1] and t ∈ (0.5, 1], let x ∈ U (λ ; t). Using the condition 3.4, we have max { λ ( k ( x - y ) + y ) , 0.5 } ≥ λ ( x ) ≥ t > 0.5 .

Thus λ (k (x - y) + y) ≥ t, and hence k (x - y) + y ⊆ U (λ ; t), that is, xy ¯ ⊆ U ( λ ; t ). Therefore the nonempty t-level set U (λ ; t) is a starshaped subset of ℝ ⁿ relative to y ∈ ℝ ⁿ for all t ∈ (0.5, 1].□

Combining Theorems 3.6 and 3.7, we have a corollary.

Corollary 3.8. For a starshaped fuzzy set λ ∈ ℱ (ℝⁿ), the nonempty t-level set U (λ ; t) is a starshaped subset of ℝ ⁿ relative to y ∈ ℝ ⁿ for all t ∈ (0, 1] if and only if λ satisfies two conditions (3.1) and (3.4).

Theorem 3.9. A fuzzy set λ ∈ ℱ (ℝⁿ) is a starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ if and only if it satisfies: ( x + y ) t ∈ λ ⇒ ( kx + y ) t ∈ ∨ q λ(3.5) for all x ∈ ℝ ⁿ , k ∈ [0, 1] and t ∈ (0, 1].

Proof. Assume that λ ∈ ℱ (ℝⁿ) is a starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ . Let k ∈ [0, 1], t ∈ (0, 1] and (x + y)  _t  ∈ λ for every x ∈ ℝ ⁿ . Then λ (x + y) ≥ t. Replacing x by x + y in (3.3),we have λ ( kx + y ) = λ ( k ( ( x + y ) - y ) + y ) = λ ( k ( x - y ) + y ) ≥ min { λ ( x ) , 0.5 } = min { λ ( x + y ) , 0.5 } ≥ min { t , 0.5 } .

If t ≤ 0.5, then λ (kx + y) ≥ t and so (kx + y)  _t  ∈ λ. If t > 0.5, then λ (kx + y) + t > 1 and so (kx + y)  _t qλ. Hence (kx + y)  _t ∈ ∨ q λ.

Conversely, suppose that λ satisfies the condition (3.4). We first show that λ ( kx + y ) ≥ min { λ ( x + y ) , 0.5 } .(3.6)

Assume that λ (x + y) <0.5. If λ (kx + y) < λ (x + y), then λ (kx + y) < t < λ (x + y) for some t ∈ (0, 0.5). Hence (x + y)  _t  ∈ λ and (kx + y)  _t oiλ. Also, since λ (kx + y) + t < 2t < 1, we get (kx + y)  _t oqλ. Thus ( kx + y ) t ∈ ∨ q ¯ λ, a contradiction. Hence λ (kx + y) ≥ λ (x + y). Now, suppose that λ (x + y) ≥0.5. Then (x + y) _0.5 ∈ λ and so (kx+y) _0.5∈ ∨ q λ by 3.5. If λ (kx + y) <0.5, then (kx + y) _0.5oiλ and λ (kx + y) +0.5 < 1, that is, (kx + y) _0.5oqλ. This is a contradiction, and so λ (kx + y) ≥0.5. Therefore λ (kx + y) ≥ min {λ (x + y) , 0.5}, Now if we replace x + y by x in (3.6), then λ ( k ( x - y ) + y ) = λ ( k ( ( x + y ) - y ) + y ) = λ ( kx + y ) ≥ min { λ ( x + y ) , 0.5 } = min { λ ( x ) , 0.5 } . It follows from Theorem 3.5 that λ is a starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ .□

Definition 3.10. A fuzzy set λ ∈ ℱ (ℝⁿ) is called a quasi-starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ if x t ∈ λ , y r ∈ λ ⇒ ( kx + ( 1 - k ) y ) min { t , r } ∈ ∨ q λ(3.7) for all x ∈ ℝ ⁿ , k ∈ [0, 1] and t, r ∈ (0, 1].

Example 3.11. The fuzzy set λ in Example 3.3 is a quasi-starshaped (∈, ∈ ∨ q)-fuzzy set relative toy = 0.

Example 3.12. The fuzzy set λ ∈ ℱ (ℝ) given by λ : ℝ → [ 0 , 1 ] , x ↦ { x + 2 if x ∈ [ - 1.5 , - 1 ) , 1 if x ∈ [ - 1 , - 0.5 ) ∪ ( 0.5 , 1 ] , x 2 if x ∈ [ - 0.5 , 0.5 ] , 2 - x if x ∈ ( 1 , 1.5 ] , 0.5 otherwise is a quasi-starshaped (∈, ∈ ∨ q)-fuzzy set relative to y = 0.

Theorem 3.13. A fuzzy set λ ∈ ℱ (ℝⁿ) is a quasi-starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ if and only if ( ∀ x ∈ ℝ n ) ( ∀ k ∈ [ 0 , 1 ] ) ( λ ( kx + ( 1 - k ) y ) ≥ min { λ ( x ) , λ ( y ) , 0.5 } )(3.8)Proof. Assume that λ is a quasi-starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ . Let x ∈ ℝ ⁿ and k ∈ [0, 1], and suppose that min {λ (x) , λ (y)} <0.5. Then λ ( kx + ( 1 - k ) y ) ≥ min { λ ( x ) , λ ( y ) } because if not then λ (kx + (1 - k) y) < t < min{λ (x) , λ (y)} for some t ∈ (0, 0.5). Thus x _t  ∈ λ andy _t  ∈ λ, but (kx + (1 - k) y) _min{t,t} = (kx + (1 - k) y)  _t oiλ and λ (kx + (1 - k) y) + t < 2t < 1, that is, (kx + (1 - k) y)  _t oqλ.

Hence ( kx + ( 1 - k ) y ) t ∈ ∨ q ¯ λ, a contradiction. Now assume that min {λ (x) , λ (y)} ≥0.5. Then x_0.5 ∈ λ and y_0.5 ∈ λ, and so (kx + (1 - k) y) _0.5∈ ∨ q λ by (3.7). Thus λ (kx + (1 - k) y) ≥0.5 because if λ (kx + (1 - k) y) <0.5 then λ (kx + (1 - k) y) +0.5 < 1, a contradiction. Consequently, λ (kx + (1 - k) y) ≥ min {λ (x) , λ (y) , 0.5} for all x ∈ ℝ ⁿ and k ∈ [0, 1].

Conversely, assume that a fuzzy set λ ∈ ℱ (ℝⁿ) satisfies the condition (3.8). Let x ∈ ℝ ⁿ , k ∈ [0, 1] and t, r ∈ (0, 1] be such that x _t  ∈ λ and y _r  ∈ λ. Then λ (x) ≥ t and λ (y) ≥ r. If λ (kx + (1 - k) y) < min {λ (x) , λ (y)}, then min {λ (x) , λ (y)} ≥0.5. Otherwise, we have λ (kx + (1 - k) y) ≥ min {λ (x) , λ (y) , 0.5} ≥ min{λ (x) , λ (y)} , a contradiction. It follows that λ ( kx + ( 1 - k ) y ) + min { t , r } > 2 λ ( kx + ( 1 - k ) y ) ≥ 2 min { λ ( x ) , λ ( y ) , 0.5 } = 1 and so that (kx + (1 - k) y) _min{t,r}qλ. Therefore λ is a quasi-starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ .□

Theorem 3.14. A fuzzy set λ ∈ ℱ (ℝⁿ) is a quasi-starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ if and only if its nonempty t-level set U (λ ; t) is starshaped relative to y ∈ ℝ ⁿ for all t ∈ (0, min {λ (y) , 0.5}].

Proof. Suppose λ is a quasi-starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ . Assume that U (λ ; t)≠ ∅ for every t ∈ (0, min {λ (y) , 0.5}]. Then y ∈ U (λ ; t), that is, λ (y) ≥ t. If x ∈ U (λ ; t), then λ (x) ≥ t. It follows from (3.8) that λ ( kx + ( 1 - k ) y ) ≥ min { λ ( x ) , λ ( y ) , 0.5 } ≥ min { t , 0.5 } = t . Hence xy ¯ ⊆ U ( λ ; t ), and so U (λ ; t) is starshaped relative to y ∈ ℝ ⁿ for all t ∈ (0, min {λ (y) , 0.5}].

Conversely, suppose the nonempty t-level set U (λ ; t) is starshaped relative to y ∈ ℝ ⁿ for all t ∈ (0, min {λ (y) , 0.5}]. For k ∈ [0, 1] and x ∈ ℝ ⁿ , let λ (y) = t _y when λ (y) < λ (x). Then xy ¯ ⊆ U ( λ ; t y ), and so λ ( kx + ( 1 - k ) y ) ≥ min { λ ( x ) , λ ( y ) } ≥ min { λ ( x ) , λ ( y ) , 0.5 } . Similarly, we have λ ( kx + ( 1 - k ) y ) ≥ min { λ ( x ) , λ ( y ) , 0.5 } by putting λ (x) = t _x when λ (x) ≤ λ (y). It follows from Theorem 3.13 that λ is a quasi-starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ .□

Theorem 3.15. Given y ∈ ℝ ⁿ , every starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ is a quasi-starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ .

Proof. Let λ be a starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ . Taking k = 0 in (3.3) induces λ (y) ≥ min {λ (x) , 0.5} for all x ∈ ℝ ⁿ . It follows from (3.3) that λ ( kx + ( 1 - k ) y ) = λ ( k ( x - y ) + y ) ≥ min { λ ( x ) , 0.5 } = min { λ ( x ) , λ ( y ) , 0.5 } for all x ∈ ℝ ⁿ and k ∈ [0, 1]. Therefore λ is a quasi-starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ by Theorem 3.13.□

Lemma 3.16. If λ ∈ ℝ ⁿ is a quasi-starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ with λ (y) ≠0.5, then the set A : = { x ∈ ℝ n ∣ λ ( x ) > 0.5 } is starshaped relative y ∈ ℝ ⁿ .

Proof. Let x ∈ A. Then λ (x) >0.5. Take t _y  = λ (y) when λ (x) > λ (y). Then, by Theorem 3.14, U (λ ; t _y ) is starshaped relative to y, and so xy ¯ ⊆ U ( λ ; t y ) ⊆ A. Similarly, if we take λ (x) = t _x when λ (x) ≤ λ (y), then xy ¯ ⊆ U ( λ ; t x ) ⊆ A. Therefore A is starshaped relative to y ∈ ℝ ⁿ .□

In Lemma 3.16, the condition λ (y) ≠0.5 is necessary. In Example 3.12, λ is a quasi-starshaped (∈, ∈ ∨ q)-fuzzy set relative to y = 2 with λ (2) =0.5. But the set A = { x ∈ ℝ n ∣ x ∈ ( - 1.5 , 0 ) } ∪ { x ∈ ℝ n ∣ x ∈ ( 0 , 1.5 ) } is not starshaped relative to y = 2.

Theorem 3.17. Let λ ∈ ℝ ⁿ be a quasi-starshaped (∈, ∈ ∨ q)-fuzzy set relative to y ∈ ℝ ⁿ with λ (y) ≠0.5. Then the closure A ¯ of A : = {x ∈ ℝ ⁿ  ∣ λ (x) >0.5} is starshaped relative to y ∈ ℝ ⁿ .

Proof. Let k ∈ [0, 1] and x 0 ∈ A ¯. Take a₀ : = kx₀ + (1 - k) y in ℝ ⁿ . Note that λ (x) = kx + (1 - k) y is continuous in x. Hence for each neighborhood G of a₀, there exists a neighborhood H of x₀ such that if x ∈ H then kx + (1 - k) y ∈ G. Since x 0 ∈ A ¯, we know that x ∈ A ∩ H. Since A is starshaped relative y by Lemma 3, we get kx + (1 - k) y ∈ A ∩ G which implies kx 0 + ( 1 - k ) y ∈ A ¯. Thus x 0 y ¯ ⊆ A ¯, and so A ¯ is starshaped relative to y ∈ ℝ ⁿ .□

Theorem 3.18. If _λ1, _λ2 ∈ ℱ (ℝⁿ) are quasi-starshaped (∈, ∈ ∨ q)-fuzzy sets relative to the point y ∈ ℝ ⁿ , then the intersection _λ1 ∩ _λ2 of _λ1 and _λ2 is also starshaped (∈, ∈ ∨ q)-fuzzy set relative to the same point.

Proof. Let k ∈ [0, 1] and x ∈ ℝ ⁿ . If _λ1, _λ2 ℱ (ℝⁿ) are quasi-starshaped (∈, ∈ ∨ q)-fuzzy sets relative to the point y ∈ ℝ ⁿ , then λ i ( kx + ( 1 - k ) y ) ≥ min { λ i ( x ) , λ i ( y ) , 0.5 } , i = 1 , 2 by Theorem 3.13. Hence ( λ 1 ∩ λ 2 ) ( kx + ( 1 - k ) y ) = min { λ 1 ( kx + ( 1 - k ) y ) , λ 2 ( kx + ( 1 - k ) y ) } ≥ min { min { λ 1 ( x ) , λ 1 ( y ) , 0.5 } , min { λ 2 ( x ) , λ 2 ( y ) , 0.5 } } = min { min { λ 1 ( x ) , λ 2 ( y ) } , min { λ 1 ( x ) , λ 2 ( y ) } , 0.5 } = min { ( λ 1 ∩ λ 2 ) ( x ) , ( λ 1 ∩ λ 2 ) ( y ) , 0.5 } , and so _λ1 ∩ _λ2 is starshaped (∈, ∈ ∨ q)-fuzzy set relative to the point y ∈ ℝ ⁿ by Theorem 3.□

Theorem 3.19. Let _λ1, _λ2 ℱ (ℝⁿ) be quasi-starshaped (∈, ∈ ∨ q)-fuzzy sets relative to the point y ∈ ℝ ⁿ . If _λ1 (y) = _λ2 (y), then the union _λ1 ∪ _λ2 of _λ1 and _λ2 is also starshaped (∈, ∈ ∨ q)-fuzzy set relative to the same point.

Proof. For any k ∈ [0, 1] and x ∈ ℝ ⁿ , let _λ1, _λ2 ℱ (ℝⁿ) be quasi-starshaped (∈, ∈ ∨ q)-fuzzy sets relative to the point y ∈ ℝ ⁿ with _λ1 (y) = _λ2 (y). Then λ i ( kx + ( 1 - k ) y ) ≥ min { λ i ( x ) , λ i ( y ) , 0.5 } , i = 1 , 2 by Theorem 3.13. Since _λ1 (y) = _λ2 (y), it follows that ( λ 1 ∪ λ 2 ) ( kx + ( 1 - k ) y ) = max { λ 1 ( kx + ( 1 - k ) y ) , λ 2 ( kx + ( 1 - k ) y ) } ≥ max { min { λ 1 ( x ) , λ 1 ( y ) , 0.5 } , min { λ 2 ( x ) , λ 2 ( y ) , 0.5 } } = min { max { λ 1 ( x ) , λ 2 ( x ) } , λ 1 ( y ) , 0.5 } = min { ( λ 1 ∪ λ 2 ) ( x ) , ( λ 1 ∪ λ 2 ) ( y ) , 0.5 } . Therefore _λ1 ∪ _λ2 is a starshaped (∈, ∈ ∨ q)-fuzzy set relative to the point y ∈ ℝ ⁿ by Theorem 3.13.□

Let Z ¯ be a generalized complex fuzzy number and assume that U ( Z ¯ ; 0.5 ) = { z 1 } and z₁ belongs to the interior of U ( Z ¯ ; α ) for α ∈ (0, 0.5) and the interior of supp ( Z ¯ ). For any α-level set U ( Z ¯ ; α ), α ∈ (0, 0.5), draw the ray L (β) from z₁ making positive angle β (β ∈ [0, 2π)) with the x-axis in the complex plane. Let Z ¯ be of star-like, that is, the intersection of L (β) and boundary of U ( Z ¯ ; α ) contains a single point, say z (α, β), for all α ∈ (0, 0.5] and β ∈ [0, 2π), and the intersection of L (β) and supp ( Z ¯ ) contains a single point, say z (0, β), for all β ∈ [0, 2π). Let z 2 ∈ U ( Z ¯ ; α ) for α ∈ (0, 0.5). Note that the ray z 1 z 2 → meets the boundary of U ( Z ¯ ; α ) at one point and U ( Z ¯ ; α ) is compact. Hence the line segment z 1 z 2 ¯ is contained in U ( Z ¯ ; α ), and so U ( Z ¯ ; α ) is starshaped relative to z₁ for α ∈ (0, 0.5). Obviously, U ( Z ¯ ; 0.5 ) = { z 1 } and U ( Z ¯ ; 0 ) = ℂ are starshaped relative to z₁. Therefore Z ¯ is a starshaped (∈, ∈ ∨ q)-fuzzy set relative to z₁.

5 Conclusion

Diamond [2] have introduced fuzzy starshaped set as a generalization of fuzzy convex fuzzy sets, and have defined metrics on the space of fuzzy starshaped sets. In this paper, we have considered a generalization of starshaped fuzzy sets, and have introduced the notion of starshaped (∈, ∈  ∨ q)-fuzzy sets and quasi-starshaped (∈, ∈  ∨ q)-fuzzy sets. We have investigated related properties, and have discussed characterizations of starshaped (∈, ∈  ∨ q)-fuzzy sets and quasi-starshaped (∈, ∈  ∨ q)-fuzzy sets. In future, we will apply this research into the other topics, for example, interval-valued (∈, ∈  ∨ q)-fuzzy set theory, rough and soft set theory, and their algebraic structures. Regarding starshaped (∈, ∈  ∨ q)-fuzzy sets and quasi-starshaped (∈, ∈  ∨ q)-fuzzy sets relative to the same point, we discuss the possibility to construct algebraic structures, for example, (distributive) lattice, residuated lattice, MV-algebra etc.