# Table 2 Definition of different types of fuzzy set [71]

Acronym Name Definition
FS Fuzzy Set A FS (or type-1 fuzzy set) A on X is a mapping A : X → [0, 1], with A(x) the membership degree of the element x of the fuzzy set A. It can also be defined as A = {(x, μA(x)) |  xX} where μA(x) : X → [0, 1].
T2FS Type-2 Fuzzy Set Is a FS for which the membership degrees are expressed as FS on [0, 1].
TnFS Type-n Fuzzy Set Is a FS whose membership values are type-(n-1) fuzzy sets.
AIFS Atanassov Intuitionistic Fuzzy Set An AIFS A on X is a mapping A : X → D([0, 1]) = {(x, y)  [0, 1]2 | x + y ≤ 1}. A(x) = (μA(x), νA(x)) for all xX. μA(x) is the membership degree of the element x to A and νA(x) is the non-membership degree. Both values should satisfy the constraint 0 ≤ μA(x) + νA(x) ≤ 1.
BVFSL Bipolar-Valued Fuzzy Set of Lee A BVFSL on X is a mapping A : X → [−1, 1]. A bipolar scale is considered, with negative values held to be opposite to positive ones.
BVFSZ Bipolar-Valued Fuzzy Set of Zhang A BVFSZ on X is a mapping A : X → [0, 1] × [−1, 0]. A(x) = (ρ+(x), ρ(x)) where ρ+: X → [0, 1], ρ : X → [−1, 0] and ρ+(x) + ρ(x) [−1, 1]. BVFSZ involves two poles on two different scales. BVFSL is a particular case of BVFSZ
CFS Complex Fuzzy Set A on the universe X is a mapping A : X → D where D = {reis | r, s [0, 1] and i =  √  − 1}. That is, the membership function of a CFS A takes its values from the unit disk on the complex plane. Therefore, FS is a particular case of CFS.
FRS Fuzzy Rough Set In a universe X, and if R is a fuzzy similarity relation on X, let AFS(X). A fuzzy rough set on X is a pair (R ↓ A, R ↑ A) FS(X) × FS(X)  where:
R ↓ A : X → [0, 1] is given by R ↓ A(x) = infuXmax (1 − R(u, x), A(u)).
R ↑ A : X → [0, 1] is given by R ↑ A(x) = supuXmin (R(u, x), A(u)).
FSS Fuzzy Soft Set A pair (F, A) is a FSS over X, where F is a mapping given by F : A → FS(X), where FS(X) denotes the set of all fuzzy subsets of X and A is a set of parameters.
GS Grey Set Defined in the same way as IVFS
HFS Hesitant Fuzzy Set This is a function that when applied to X returns a subset of [0, 1].
THFS Typical Hesitant Fuzzy Set This is a HFS where the membership degree of each of the elements is given by a finite and non-empty subset of [0, 1].
IT2FS Interval Type-2 Fuzzy Set When all μA(x, u) = 1, then A is an IT2FS corresponding to A(x) = {(u, 1)| uJx [0, 1]} for every xX.
IVAIFS Interval-Valued Atanassov Intuitionistic Fuzzy Set If A on X is a mapping $$A:X\to LL\left(\left[0,1\right]\right)=\left\{\left(\left[\underset{\_}{\mu },\overline{\mu}\right],\left[\underset{\_}{\nu },\overline{\nu},\Big\}\right]\right)|\left[\underset{\_}{\mu },\overline{\mu}\right],\left[\underset{\_}{\nu },\overline{\nu},\right]\in L\left(\left[0,1\right]\right)\ such\ that\ \overline{\mu}+\overline{\nu}\le 1\right\}$$. Where L([0, 1]) denotes the set of all closed subintervals of the unit interval.
IVFS Interval-Valued Fuzzy Set If L([0, 1]) is the set of all closed subintervals of [0, 1], i.e., $$L\left(\left[0,1\right]\right)=\left\{\left[\underset{\_}{x},\overline{x}\right]|\left(\underset{\_}{x},\overline{x}\right)\in {\left[0,1\right]}^2\ \mathrm{and}\ \underset{\_}{x}\le \overline{x}\right\}$$. An IVFS A on X is a mapping A : X → L([0, 1]).
mPVFS m-Polar-Valued Fuzzy Set If m ≥ 2. An mPVFS on X is a mapping A : X → {1, …, m} × [0, 1]
NS Neutrosophic Set A NS A on X is a mapping A : X → [0, 1]2.
PFS Pythagorean Fuzzy Set A PFS A on X is a mapping A : X → {(x, y)  [0, 1]2 |  x2 + y2 ≤ 1}.
SVFS Set-Valued Fuzzy Set This is a FS for which membership degrees are expressed as subsets on [0, 1]. It can also be defined as A on X as a mapping A : X → 2[0, 1]\{}, where 2[0, 1] is the power set of [0, 1], that is, the set of all subsets of [0, 1].
SS Shadow Set Given AFS(X), a shadow set B induced by A is an IVFS on X such that the membership degree of an element xX is either [0, 0], [1, 1] or [0, 1];
VS Vague Set Introduced by Gau and Buehrer in 1993. In 1994, it was shown to be the same as AIFS.