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单项选择题设事件A，B，C满足ABC≠，若将事件A∪B∪C表示成互不相容事件之和，则下列表示方法错误的是
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【并集】一般地，由属于集合A或者属于集合B的所有元素组成的集合，称为A与B的并集（union&set），记作A∪B（读作“&A&并&B&”'），即&A∪B=\left\{{x\left|{x∈A,或x∈B}\right}\right\}．&&【交集】一般地，由属于集合A并且属于集合B的所有元素组成的集合，称为A与B的交集（intersection&set），记作&A∩B（读作“&A&交&B&”'），即&A∩B\left\{{=x\left|{x∈A,且x∈B}\right}\right\}．&&【补集】一般地，如果一个集合含有我们所研究问题中涉及的所有对象，那么就称这个集合为全集（universe&set），通常记为U．对于一个集合&A，由全集U中不属于集合A的所有元素组成的集合称为集合A相对于全集U的补集（complementary&set），简称为集合A的补集（补），记作{{C}_{U}}A．
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试题“已知A与B是集合{1，2，3，…，100}的两个子集，满足：...”，相似的试题还有：
设全集U={1，2，3，4，5}，A∩C∪B={1，2}，则集合C∪A∩B的所有子集个数最多为()
定义集合运算A×B={(a，b)|a∈A∩B，b∈A∪B}，若A={0，1}，B={1，2}，则A×B的子集有()
已知集合A={x|(x-2)[x-(3a+1)]＜0}，集合B=．(1)当a=2时，求A∩B；(2)当a时，若元素x∈A是x∈B的必要条件，求实数a的取值范围．a_甜梦文库
An Outline of a Theory of Three-way DecisionsYiyu YaoDepartment of Computer Science, University of Regina, Regina, Saskatchewan, Canada S4S 0A2 E-mail: yyao@cs.uregina.caAbstract. A theory of three-way decisions is constructed based on the notions of acceptance, rejection and noncommitment. It is an extension of the commonly used binary-decision model with an added third option. Three-way decisions play a key role in everyday decision-making and have been widely used in many ?elds and disciplines. An outline of a theory of three-way decisions is presented by examining its basic ingredients, interpretations, and relationships to other theories.1IntroductionThe concept of three-way decisions was recently proposed and used to interpret rough set three regions [52, 54, 55]. More speci?cally, the positive, negative and boundary regions are viewed, respectively, as the regions of acceptance, rejection, and noncommitment in a ternary classi?cation. The positive and negative regions can be used to induce rules of acce whenever it is impossible to make an acceptance or a rejection decision, the third noncommitement decision is made [54]. It can be shown that, under certain conditions, probabilistic three-way decisions are superior to both Palwak three-way decisions and two-way (i.e., binary) decisions [55]. Many recent studies further investigated extensions and applications of three-way decisions [1, 7C10, 12, 13, 17C21, 23C29, 31, 45, 46, 56, 60C62, 64C66]. The essential ideas of three-way decisions are commonly used in everyday life [32] and widely applied in many ?elds and disciplines, including, for example, medical decision-making [30, 37, 38], social judgement theory [39], hypothesis testing in statistics [42], management sciences [5, 44], and peering review process [43]. However, a close examination surprisingly reveals that there still does not exist a uni?ed formal description. To extend the concept of three-way decisions of rough sets to a much wider context, this paper outlines a theory of three-way decisions.Information about this paper: Yao, Y.Y. An outline of a theory of threeway decisions. In: Yao, J., Yang, Y., Slowinski, R., Greco, S., Li, H., Mitra, S., Polkowski, L. (eds.) RSCTC 2012. LNCS (LNAI), vol. 7413, pp. 1-17. Springer, Heidelberg (2012)2A Description of Three-way DecisionsThe essential ideas of three-way decisions are described in terms of a ternary classi?cation according to evaluations of a set of criteria. Suppose U is a ?nite nonempty set of objects or decision alternatives and C is a ?nite set of conditions. Each condition in C may be a criterion, an objective, or a constraint. For simplicity, in this paper we refer to conditions in C as criteria. Our decision task is to classify objects of U according to whether they satisfy the set of criteria. In widely used two-way decision models, it is assumed that an object either satis?es the criteria or does not satisfy the criteria. The set U is divided into two disjoint regions, namely, the positive region POS for objects satisfying the criteria and the negative region NEG for objects not satisfying the criteria. There are usually some classi?cation errors associated with such a binary classi?cation. Two main di?culties with two-way approaches are their stringent binary assumption of the satis?ability of objects and the requirement of a dichotomous classi?cation. In many situations, it may happen that an object only satis?es the set of criteria to some degree. Even if an object may actually either satisfy or not satisfy the criteria, we may not be able to identify without uncertainty the subset of objects that satisfy the criteria due to uncertain or incomplete information. Consequently, we are only able to search for an approximate solution. Instead of making a binary decision, we use thresholds on the degrees of satis?ability to make one of three decisions: (a) accept an object as satisfying the set of criteria if its degree of satis?ability is at or a (b) reject the object by treating it as not satisfying the criteria if its degree of satis?ability is at or and (c) neither accept nor reject the object but opt for a noncommitment. The third option may also be referred to as a deferment decision that requires further information or investigation. From the informal description, we give a formal de?nition. The problem of three-way decisions. Suppose U is a ?nite nonempty set and C is a ?nite set of criteria. The problem of three-way decisions is to divide, based on the set of criteria C , U into three pair-wise disjoint regions, POS, NEG, and BND, called the positive, negative, and boundary regions, respectively. Corresponding to the three regions, one may construct rules for three-way decisions. In our previous studies [52, 54], we used three types of rules, namely, rules for acceptance, rejection, and noncommitment, respectively. It now appears to us that only rules for acceptance and rules for rejection are meaningful and su?cient. That is, the noncommitment set is formed by those objects to which neither a rule for acceptance nor a rule for rejection applies. It is not necessary to have, and in many cases may be impossible to construct, rules for noncommitement. To formally describe the satis?ability of objects, rules for acceptance and rules for rejection, we need to introduce the notion of evaluations of objects and 2designated values for acceptance and designated values for rejection. Evaluations provide the degrees of satis?ability, designated values for acceptance are acceptable degrees of satis?ability, and designated valued for rejection are acceptable degrees of non-satis?ability. They provide a basis for a theory of three-way decisions. A theory of three-way decisions must consider at least the following three issues regarding evaluations and designated values: 1. Construction and interpretation of a set of values for measuring satis?ability and a set of values for measuring non-satis?ability. The former is used by an evaluation for acceptance and the latter is used by an evaluation for rejection. In many cases, a single set may be used by both. It is assumed that the set of evaluation values is equipped with an ordering relation so that we can compare at least some objects according to their degrees of satis?ability or non-satis?ability. Examples of a set evaluation values are a poset, a lattice, a set of a ?nite numbers of grades, the set of integers, the unit interval, and the set of reals. Social judgement theory uses latitudes of acceptance, rejection, and noncommitment [6, 39], which is closely related to our formulation of three-way decisions. 2. Construction and interpretation of evaluations. An evaluation depends on the set of criteria and characterizes either satis?ability or non-satis?ability of objects in U . Evaluations for the purposes of acceptance and rejection may be either independent or the same. Depending on particular applications, evaluations may be constructed and interpreted in terms of more intuitive and practically operable notions, including costs, risks, errors, pro?ts, bene?ts, user satisfaction, committee voting, and so on. Based on the values of an evaluation, one can at least compare some objects. 3. Determination and interpretation of designated values for acceptance and designated values for rejection. The sets of designated values must meaningfully re?ect an intuitive understanding of acceptance and rejection. For example, we can not accept and reject an object simultaneously. This requires that the set of designated values for acceptance and the set of designated value for rejection are disjoint. The designated values for acceptance should lead to if we accept an object x then we should accept all those objects that have the same or larger degrees of satis?ability than x. It is also desirable if we can systematically determine the sets of designated values on a semantically sound basis. By focusing on these issues, we examine three classes of evaluations. Evaluations are treated as a primitive notion for characterizing the satis?ability or desirability of objects. Their concrete physical interpretations are left to particular applications. 33Evaluation-based Three-way DecisionsWe assume that evaluations for acceptance and rejection can be constructed based on the set of criteria. This enables us to focus mainly on how to obtain three-way decisions according to evaluations. The problem of constructing and interpreting evaluations is left to further studies and speci?c applications. A framework of evaluation-based three-way decisions is proposed and three models are introduced and studied. 3.1 Three-way Decisions with a Pair of Poset-based EvaluationsFor the most general case, we consider a pair of (may be independent) evaluations, one for the purpose of acceptance and the other for rejection. De?nition 1. Suppose U is a ?nite nonempty set and (La , a ) (Lr , r ) are two posets. A pair of functions va : U ?→ La and vr : U ?→ Lr is called an acceptance evaluation and a rejection evaluation, respectively. For x ∈ U , va (x) and vr (x) are called the acceptance and rejection values of x, respectively. In real applications, the set of possible values of acceptance may be interpreted based on more operational notions such as our con?dence of an object satisfying the given set of criteria, or cost, bene?t, and value induced by the object. For two objects x, y ∈ U , if va (x) a va (y ), we say that x is less acceptable than y . By adopting a poset (La , a ), we assume that some objects in U are incomparable. Similar interpretation can be said about the possible values of an evaluation for rejection. In general, acceptance and rejection evaluations may be independent. To accept an object, its value va (x) must be in a certain subset of La representing the acceptance region of La . Similarly, we need to de?ne the rejection region of Lr . By adopting a similar terminology of designated values in manyvalued logics [4], these values are called designated values for acceptance and designated values for rejection, respectively. Based on the two sets of designated values, one can easily obtain three regions for three-way decisions. De?nition 2. Let ? = L+ a ? La be a subset of La called the designated values for acceptance, and ? = L? r ? Lr be a subset of Lr called the designated values for rejection. The positive, negative, and boundary regions of three-way decisions induced by (va , vr ) are de?ned by:+ ? ? (va , vr ) = {x ∈ U | va (x) ∈ L POS(L+ a ∧ vr (x) ∈ Lr }, a ,Lr ) + ? ? (va , vr ) = {x ∈ U | va (x) ∈ L NEG(L+ a ∧ vr (x) ∈ Lr }, a ,Lr ) c ? (va , vr ) = (POS ? (va , vr ) ∪ NEG ? (va , vr )) BND(L+ (L+ (L+ a ,Lr ) a ,Lr ) a ,Lr ) ? = {x ∈ U | (va (x) ∈ L+ a ∧ vr (x) ∈ Lr ) ∨ ? (va (x) ∈ L+ a ∧ vr (x) ∈ Lr )}.(1)4The boundary region is de?ned as the complement of the union of positive and negative regions. The conditions in the de?nition of the positive and negative regions make sure that they are disjoint. Therefore, the three regions are pairwise disjoint. The three regions do not necessarily form a partition of U , as some of them may be empty. In fact, two-way decisions may be viewed as a special case of three-way decisions in which the boundary region is always empty. By the interpretation of the orderings a and r , the designated values L+ a for acceptance and the designated values L? r for rejection must satisfy certain + properties. If La has the largest element 1, then 1 ∈ L+ a u and w ∈ La , a . If w + then u ∈ La . That is, if va (x) a va (y ) and we accept x, then we must accept y . ? Similarly, if Lr has the largest element 1, then 1 ∈ L? r u and w ∈ Lr , r . If w ? then u ∈ Lr . 3.2 Three-way Decisions with One Poset-based EvaluationIn some situations, it may be more convenient to combine the two evaluation into a single acceptance-rejection evaluation. In this case, one poset (L, ) is used and two subsets of the poset are used as the designated values for acceptance and rejection, respectively. De?nition 3. Suppose (L, ) is a poset. A function v : U ?→ L is called an acceptance-rejection evaluation. Let L+ , L? ? L be two subsets of L with L+ ∩ L? = ?, called the designated values for acceptance and the designated values for rejection, rspectively. The positive, negative, and boundary regions of three-way decisions induced by v is de?ned by: POS(L+ ,L? ) (v ) = {x ∈ U | v (x) ∈ L+ }, NEG(L+ ,L? ) (v ) = {x ∈ U | v (x) ∈ L? }, BND(L+ ,L? ) (v ) = {x ∈ U | v (x) ∈ L+ ∧ v (x) ∈ L? }. (2) The condition L+ ∩ L? = ? ensures that the three regions are pair-wise disjoint. A single evaluation v may be viewed as a special case of two evaluations in which a = and r = . In this way, acceptance is related to rejection in the sense that the reverse ordering of acceptance is the ordering for rejection. To ensure the meaningfulness of L+ and L? , it is required that ?(w u) for all w ∈ L+ and u ∈ L? . In other words, L+ contains larger elements of L and L? contains smaller elements of L. 3.3 Three-way Decisions with an Evaluation Using a Totally Ordered SetConsider now an evaluation based on a totally ordered set (L, ) where is a total order. That is, is a partial order and any two elements of L are comparable. This is in fact a widely used approach. For example, L is either the set of real numbers or the unit interval [0, 1] and is the less-than-or-equal relation ≤. For a total order, it is possible to de?ne the sets of designated values for acceptance and rejection by a pair of thresholds. 5De?nition 4. Suppose (L, ) is a totally ordered set, that is, is a total order. For two elements α, β with β ? α (i.e., β α ∧ ?(α β )), suppose that the set of designated values for acceptance is given by L+ = {t ∈ L | t α} and the set of designated values for rejection is given by L? = {b ∈ L | b β }. For an evaluation function v : U ?→ L, its three regions are de?ned by: POS(α,β ) (v ) = {x ∈ U | v (x) NEG(α,β ) (v ) = {x ∈ U | v (x) α}, β }, (3)BND(α,β ) (v ) = {x ∈ U | β ? v (x) ? α}.Although evaluations based on a total order are restrictive, they have a computational advantage. One can obtain the three regions by simply comparing the evaluation value with a pair of thresholds. It is therefore not surprising to ?nd that many studies in fact use a total order. 3.4 Comments on Evaluations and Designated ValuesConstruction and interpretation of evaluations and designated values are vital for practical applications of three-way decisions. At a theoretical level, it may be only possible to discuss required properties of evaluations. It is assumed that an evaluation is determined by a set of criteria, representing costs, bene?ts, degrees of desirability, objectives, constraints, and so on. Further studies on evaluations may be a fruitful research direction. As an illustration, consider a simple linear model for constructing an evaluation. Suppose C = {c1 , c2 , . . . , cm } are a set of m criteria. Suppose vci : U ?→ denotes an evaluation based on criterion vi , 1 ≤ i ≤ m. An overall evaluation function v : U ?→ may be simply de?ned by a linear combination of individual evaluations: v (x) = vc1 (x) + vc2 (x) + . . . + vcm (x). (4) Details of this linear utility model and other models can be found in literature of multi-crieria and multi-objective decision making [14]. Construction and interpretation of designated values may be explained in terms of bene?ts or risks of the resulting three regions of three-way decisions. For example, consider the model that uses a total order. Let RP (α, β ), RN (α, β ) and RB (α, β ) denote the risks of the positive, negative, and boundary regions, respectively. It is reasonable to require that the sets of designated values are chosen to minimize the following overall risks: R(α, β ) = RP (α, β ) + RN (α, β ) + RB (α, β ). (5)That is, ?nding a pair of thresholds can be formulated as the following optimization problem: arg min R(α, β ). (6)(α,β )As a concerte example, R may be understood as uncertainty associated with three regions, by minimizing the overall uncertainty one can obtain the set of 6designed values in a probabilistic rough set model [2]. Two additional examples will be given in the next section when reviewing decision-theoretic rough sets [50, 57, 58] and shadowed sets [34, 35].4Models of Three-way DecisionsWe show that many studies on three-way decisions can be formulated within the framework proposed in the last section. For simplicity and as examples, we focus on the concept of concepts in a set-theoretical setting. In the classical view of concepts [40, 41], every concept is understood as a unit of thought consisting of two parts, the intension and the extension of the concept. Due to uncertain or insu?cient information, it is not always possible to precisely have a set of objects as the extension of a concept. Consequently, many generalizations of sets have been proposed and studied. 4.1 Interval Sets and Three-valued LogicInterval sets provide a means to describe partially known concepts [47, 53]. On the one hand, it is assumed that an object may actually be either an instance or not an instance of a concept. On the other hand, due to a lack of information and knowledge, one can only express the state of instance and non-instance for some objects, instead of all objects. That is, one has a partially known concept de?ned by a lower bound and upper bound of its extension. Formally, a closed interval set is a subset of 2U of the form, [Al , Au ] = {A ∈ 2U | Al ? A ? Au }, (7)where it is assumed that Al ? Au , and Al and Au are called the lower and upper bound, respectively. Any set X ∈ [Al , Au ] may be the actual extension of the partially known concept. Constructive methods for de?ning interval sets can be formulated within an incomplete information table [16, 22]. An interval set is an interval of the power set lattice 2U ; it is also a lattice, with the minimum element Al , the maximum element Au , and the standard set-theoretic operations. Interval-set algebra is related to Kleene’s three-valued logic [15, 36], in which a third truth value is added to the standard two-valued logic. The third value may be interpreted as unknown or undeterminable. Let L = {F, I, T } denote the set of truth values with a total order F I T . An interval set [Al , Au ] can be equivalently de?ned by an acceptance-rejection evaluation as, ? x ∈ (Au )c , ? F, x ∈ Au ? Al , v[Al ,Au ] (x) = I, (8) ? T, x ∈ Al . Suppose the sets of designated values for acceptance and rejection are de?ned by a pair of thresholds (T, F ), namely, L+ = {a ∈ L | T a} = {T } and 7L? = {b ∈ L | b F } = {F }. According to De?nition 4, an interval set provides the following three-way decisions: POS(T,F ) ([Al , Au ]) = {x ∈ U | v[Al ,Au ] (x) NEG(T,F ) ([Al , Au ]) = {x ∈ U | v[Al ,Au ] (x) T } = Al , F } = (Au )c , (9)BND(T,F ) ([Al , Au ]) = {x ∈ U | F ? v[Al ,Au ] (x) ? T } = Au ? Al .Although the re-expression of an interval set in terms of three-way decisions is somewhat trivial, it does provide a new view to look at interval sets. 4.2 Pawlak Rough SetsPawlak rough set theory deals with approximations of a concept based on a family of de?nable concepts [33]. Let E ? U × U denote an equivalence relation on U , that is, E is re?exive, symmetric, and transitive. The equivalence class containing x is de?ned by [x]E = [x] = {y ∈ U | xEy }, which is a set of objects equivalent to x. The family of all equivalence classes of E is called the quotient set induced by E , denoted as U/E . In an information table, an equivalence class is a de?nable set that can be de?ned by the conjunction of a family of attribute-value pairs [49]. For a subset A ? U , the Pawlak rough set lower and upper approximations of A are de?ned by: apr(A) = {x ∈ U | [x] ? A}, apr(A) = {x ∈ U | [x] ∩ A = ?} = {x ∈ U | ?([x] ? Ac )}. (10) In the de?nition, we use an equivalent condition ?([x] ? Ac ) so that both lower and upper approximations are de?ned uniformly by using set inclusion ?. According to the pair of approximations, the Pawlak positive, negative and boundary regions are de?ned by: POS(A) = apr(A), = {x ∈ U | [x] ? A}; NEG(A) = U ? apr(A), = {x ∈ U | [x] ? Ac }; BND(A) = apr(A) ? apr(A), = {x ∈ U | ?([x] ? Ac ) ∧ ?([x] ? A)} = (POS(A) ∪ NEG(A))c . (11)Again, these regions are de?ned uniformly by using set inclusion. The three regions are pair-wise disjoint. Conversely, from the three regions, we can compute the pair of approximations by: apr(A) = POS(A) apr(A) = POS(A) ∪ BND(A). 8Therefore, rough set theory can be formulated by either a pair of approximations or three regions. Three-way decisions with rough sets can be formulated as follows. Let La = ? Lr = {F, T } with F T , and let L+ a = Lr = {T }. All objects in the same equivalence class have the same description. Based on descriptions of objects, we have a pair of an acceptance evaluation and a rejection evaluation: ? ? [x] ? A, [ x ] ? Ac , ? T, ? T, v(a,A) (x) = v(r,A) (x) = (12) ? ? F, ?([x] ? A); F, ?([x] ? Ac ). According to De?nition 2, for a set A ? U , we can make the following three-way decisions: POS({T },{T }) (A) = {x ∈ U | v(a,A) (x) ∈ {T } ∧ v(r,A) (x) ∈ {T }} = {x ∈ U | v(a,A) (x) = T } = {x ∈ U | [x] ? A}, NEG({T },{T }) (A) = {x ∈ U | v(a,A) (x) ∈ {T } ∧ v(r,A) (x) ∈ {T }}, = {x ∈ U | v(r,A) (x) = T } = {x ∈ U | [x] ? Ac }, BND({T },{T }) (A) = (POS(va , vr ) ∪ NEG(va , vr ))c = {x ∈ U | ?([x] ? A) ∧ ?([x] ? Ac )}. (13)The reformulation of rough set three regions based uniformly on set inclusion provides additional insights into rough set approximations. It explicitly shows that acceptance is based on an evaluation of the condition [x] ? A and rejection is based on an evaluation of the condition [x] ? Ac . By those two conditions, both decisions of acceptance and rejection are made without any error. Whenever there is any doubt, namely, ?([x] ? A) ∧ ?([x] ? Ac ), a decision of noncommitment is made. 4.3 Decision-Theoretic Rough SetsDecision-theoretic rough sets (DTRS) [48, 50, 51, 57, 58] are a quantitative generalization of Pawlak rough sets by considering the degree of inclusion of an equivalence class in a set. The acceptance-rejection evaluation used by a DTRS model is the conditional probability vA (x) = P r(A|[x]), with values from the totally ordered set ([0, 1], ≤). Given a pair of thresholds (α, β ) with 0 ≤ β & α ≤ 1, the sets of designated values for acceptance and rejections are L+ = {a ∈ [0, 1] | α ≤ a} and L? = {b ∈ [0, 1] | b ≤ β }. According to De?nition 4, a DTRS model makes the following three-way decisions: for A ? U , POS(α,β ) (A) = {x ∈ U | vA (x) α}= {x ∈ U | P r(A|[x]) ≥ α}, 9NEG(α,β ) (A) = {x ∈ U | vA (x)β}= {x ∈ U | P r(A|[x]) ≤ β }, BND(α,β ) (A) = {x ∈ U | β ? vA (x) ? α} = {x ∈ U | β & P r(A|[x]) & α}. (14) Three-way decision-making in DTRS can be easily related to incorrect acceptance error and incorrect rejection error [55]. Speci?cally, incorrect acceptance error is given by P r(Ac |[x]) = 1 ? P r(A|[x]) ≤ 1 ? α, which is bounded by 1 ? α. Likewise, incorrect rejection error is given by P r(A|[x]) ≤ β , which is bounded by β . Therefore, the pair of thresholds can be interpreted as de?ning tolerance levels of errors. A main advantage of a DTRS model is its solid foundation based on Bayesian decision theory. In addition, the pair of thresholds can be systematically computed by minimizing overall ternary classi?cation cost [55]. Bayesian decision theory [3] can be applied to the derivation of DTRS as follows. We have a set of 2 states and a set of 3 actions for each state. The set of states is given by ? = {A, Ac } indicating that an object is in A and not in A, respectively. For simplicity, we use the same symbol to denote both a subset A and the corresponding state. With respect to the three regions, the set of actions with respect to a state is given by A = {aP , aN , aB }, where aP , aN , and aB represent the three actions in classifying an object x, namely, deciding x ∈ POS(A), deciding x ∈ NEG(A), and deciding x ∈ BND(A), respectively. The losses regarding the risk or cost of those classi?cation actions with respect to di?erent states are given by the 3 × 2 matrix: aP aN aB A (P ) Ac (N ) λPP λPN λNP λNN λBP λBNIn the matrix, λPP , λNP and λBP denote the losses incurred for taking actions aP , aN and aB , respectively, when an object belongs to A, and λPN , λNN and λBN denote the losses incurred for taking the same actions when the object does not belong to A To determine a pair of thresholds for three-way decisions, one can minimize the following overall risk [12, 55]: R(α, β ) = RP (α, β ) + RN (α, β ) + RB (α, β ), where RP (α, β ) = RN (α, β ) =P r (A|[x])≤β(15)[λPP P r(A|[x]) + λPN P r(Ac |[x])]P r([x]),P r (A|[x])≥α[λNP P r(A|[x]) + λNN P r(Ac |[x])]P r([x]), [λBP P r(A|[x]) + λBN P r(Ac |[x])]P r([x]),β&P r (A|[x])&αRB (α, β ) =(16)10represent, risks incurred by acceptance, rejection, and noncommitment, and the summation is over all equivalence classes. It can be shown [50, 55] that under the following conditions, (c1 ) (c2 ) λPP & λBP & λNP , λNN & λBN & λPN , (17)(λPN ? λBN )(λNP ? λBP ) & (λBN ? λNN )(λBP ? λPP ),a pair of threshold (α, β ) with 0 ≤ β & α ≤ 1 that minimizes R is given by: α= (λPN ? λBN ) , (λPN ? λBN ) + (λBP ? λPP ) (λBN ? λNN ) . β= (λBN ? λNN ) + (λNP ? λBP )(18)That is, the pair of thresholds can be computed from the loss function. Other models for determining the pair of thresholds include a game-theoretic framework [1, 9, 11], a multi-view decision model [17, 66], and the minimization of uncertainty of the three regions [2]. The conditional probability required by DTRS can be estimated based on a naive Bayesian rough set model [59] or a regression model [25]. 4.4 Three-valued Approximations in Many-valued Logic and Fuzzy SetsThree-valued approximations in many-valued logics are formulated based on the discussion given by Gottwald [4] on positively designated truth degrees and negatively designated truth degrees. In many-valued logic, the set of truth degrees or values is normally an ordered set (L, ) and contains the classical truth values F and T (often coded by 0 and 1) as its minimum and maximum elements, namely, {F, T } ? L and for any u ∈ L, F u T . It is also a common practice to use a subset L+ of positively designated truth degrees to code the intuitive notion of truth and to use another subset L? of negatively designated truth degrees to code the opposite. For the two sets to be meaningful, the following conditions are normally assumed [4]: (i) (ii) (iii) T ∈ L+ , F ∈ L? , L+ ∪ L? ? L, L+ ∩ L? = ?, w u ∧ w ∈ L+ =? u ∈ L+ , w u ∧ u ∈ L? =? w ∈ L? .Three-valued approximations of a many-valued logic derive from three-way decisions based on the two designated sets. We accept a truth degree as being true if it is in the positively designated set, reject it as being true if it is the negatively 11designated set, and neither accept nor reject if it is not in any of the two sets. By so doing, we can have a new three-valued logic with the set of truth values L3 = {L? , L ? (L? ∪ L+ ), L+ } under the ordering L? 3 L ? (L? ∪ L+ ) 3 L+ , which is an approximation of the many-valued logic. To a large extent, our formulation of three-way decisions, as given in the last section, draws mainly from such a consideration. Speci?cally, we borrowed the notions of designated truth degrees from studies of many-valued logic to introduce the notions of designated values for acceptance and rejection in the theory of three-way decisions. A fuzzy set A is characterized by a mapping from U to the unit interval, namely, ?A : U ?→ [0, 1]. The value ?A (x) is called the degree of membership of the object x ∈ U . Fuzzy sets may be interpreted in terms of a many-valued logic with the unit interval as its set of truth degrees. According to the three-valued approximations of a many-valued logic, one can similarly formulate three-valued approximations of a fuzzy set. This formulation was in fact given by Zadeh [63] in his seminal paper on fuzzy sets and was shown to be related to Kleene’s three-valued logic. Given a pair of thresholds (α, β ) with 0 ≤ β & α ≤ 1, one can de?ne the designated sets of values for acceptance and rejection as L+ = {a ∈ [0, 1] | α ≤ a} and L? = {b ∈ [0, 1] | b ≤ β }. According to De?nition 4, if a fuzzy membership function ?A is used as an acceptance-rejection evaluation, namely, v?A = ?A , we have the following three-way decisions, POS(α,β ) (?A ) = {x ∈ U | v?A (x) NEG(α,β ) (?A ) = {x ∈ U | v?A (x) α} β}= {x ∈ U | ?A (x) ≥ α}, = {x ∈ U | ?A (x) ≤ β }, BND(α,β ) (?A ) = {x ∈ U | β ? v?A (x) ? α} = { x ∈ U | β & ? A ( x) & α } . (19)Zadeh [63] provided an interpretation of this three-valued approximations of a fuzzy set: one may say that (1) x belongs to A if ?A (x) ≥ α; (2) x does not belong to A if ?A (x) ≤ β ; and (3) x has an indeterminate status relative to A if β & ?A (x) & α. This interpretation explicitly uses the notions of acceptance and rejection and is consistent with our three-way decisions. 4.5 Shadowed SetsIn contrast to decision-theoretic rough sets in which the pair of thresholds can be interpreted by classi?cation errors, there is a di?culty in interpreting thresholds in three-valued approximations of a fuzzy sets. The introduction of a shadowed set induced by a fuzzy set attempts to address this problem [34, 35]. A shadowed set A is de?ned as a mapping, SA : U ?→ {0, [0, 1], 1}, from U to a set of three truth values. It is assumed that the three values are ordered by 0 [0, 1] 1. The value [0, 1] represents the membership of objects in the 12shadows of a shadowed set. Like the interval-set algebra, shadowed-set algebra is also related to Kleene’s three-valed logic. Shadowed sets provide another model of three-way decisions. Unlike an interval set, a shadowed set is constructed from a fuzzy set ?A : U ?→ [0, 1] as follows: ? ?A (x) ≤ τ, ? 0, [0, 1], τ & ?A (x) & 1 ? τ, SA (x) = (20) ? 1, ?A (x) ≥ 1 ? τ, where 0 ≤ τ & 0.5 is a threshold. Given a pair of thresholds (1, 0) for the set of truth values {0, [0, 1], 1}, by De?nition 4 and equations (19) and (20), we have the following three-way decision for a shadowed set: POS(1,0) (SA ) = {x ∈ U | vSA (x) = POS(1?τ,τ ) (?A ), NEG(1,0) (SA ) = {x ∈ U | vSA (x) = NEG(1?τ,τ ) (?A ), BND(1,0) (SA ) = {x ∈ U | 0 ? vSA (x) ? 1} = {x ∈ U | τ & ?A (x) & 1 ? τ } = BND(1?τ,τ ) (?A ). (21) That is, a shadowed set is a three-valued approximation of a fuzzy set with (α, β ) = (1 ? τ, τ ). In general, one can also consider shadowed set by a pair of thresholds (α, β ) with 0 ≤ β & α ≤ 1 on a fuzzy set ?A . As shown in [34, 35], the threshold τ for constructing a shadowed set can be determined by minimizing the following function, ? (τ ) = abs(?r (τ ) + ?e (τ ) ? ?s (τ )), where abs(?) stands for the absolute value and ?r (τ ) ={x∈U |?A (x)≤τ }1}= {x ∈ U | ?A (x) ≥ 1 ? τ } 0}= {x ∈ U | ?A (x) ≤ τ },(22)?A (x), (1 ? ?A (y )),{y ∈U |?A (y )≥1?τ }?e (τ ) =?s (τ ) = card({z ∈ U | τ & ?A (z ) & 1 ? τ }),(23)are, respectively, the total of reduced membership values from ?A (x) in the fuzzy set to 0 in the shadowed set (i.e., ?A (x) ? 0 = ?A (x)), the total of elevated membership values from ?A (y ) in the fuzzy set to 1 in the shadowed set (i.e., 1 ? ?A (y )), and the cardinality of the shadows of the shadowed set. The minimization of ? (τ ) may be equivalently formulated as ?nding a solution to the equation, ?r (τ ) + ?e (τ ) = ?s (τ ), 13 (24)if it has a solution. Although the problem of ?nding the threshold τ is formulated precisely, the meaning of the objective function ? (τ ) still needs further investigation. It is interesting to note that the objective function ? (τ ) shares some similarity to the objective function R(α, β ) of a DTRS model, which may shed some light on the problem of determining the threshold in shadowed sets.5ConclusionsThe concept of three-way decisions provides an appealing interpretation of three regions in probabilistic rough sets. The positive and negative regions are sets of accepted objects and rejected objects, respectively. The boundary region is the set of objects for which neither acceptance nor rejection is possible, due to uncertain or incomplete information. A close examination of studies and applications of three-way decisions shows that (a) essential ideas of three-way decisions are general applicable to a wide range of decision- (b) we routinely make three-way decisi (c) three-way decisions appear across many ? and (d) there is a lack of formal theory for three-way decisions. These ?ndings motivate a study of a theory of three-way decisions in its own right. 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