The coronavirus disease 2019 (COVID-19) pandemic has disrupted the original offline teaching mode in colleges and universities. Under the background of “suspending classes without stopping teaching, suspending classes without stopping learning,” we developed an online teaching mode of calculus courses using the Superstar network teaching platform and successfully completed the online teaching process including lesson preparation, teaching, and examinations. We found some problems in the system during online teaching, for example, in “brush the lesson” learning [1,2]. We know that some students just wanted to complete their study tasks; they opened the course videos and did other things after signing in. Some students used the so-called “lesson brush artifact” to complete their study tasks. We also found that there are some problems in the online examinations; for example, as there are no monitors, some students may copy the answers or cheat in other ways. Therefore, it is important to evaluate online learning and student satisfaction. In [3–5], the authors evaluated the impact of shifting from traditional learning to online learning during the COVID-19 pandemic on undergraduate students and examined the positive and negative aspects of online learning from the students’ perspectives. Thom et al. [6] examined the lessons learned in online learning during the COVID-19 pandemic. However, it is also necessary to evaluate the credibility of online examinations. In this study, we investigated the credibility of the online examination for the calculus course we had developed. According to pedagogy theory [7], students’ scores are related to the following factors: intelligence level, course basis, psychological state, learning method, and learning time. In particular, we focused on the following four aspects: students’ proficiency in the course, appropriate learning methods, degree of class attendance, and completion of homework.

To investigate the credibility of the online examination, we randomly selected 30 engineering students at Wenzheng College of Soochow University as a sample and established an information system S = (U, C ∪ D, f). Online learning data for these students are presented in Table 1. We used the rough-set theory to analyze Table 1 and extract useful information hidden in these data. Rough-set theory, which is a logic-mathematical method proposed by Pawlak [8,9], is an effective tool for extracting and analyzing useful information hidden in data. In recent years, the rough-set theory has been widely implemented in many fields of natural and social sciences [10–14]. In 2009, Yao [15,16] introduced a third decision, the boundary decision, based on a two-way decision and proposed the three-way decision theory. The three-way decision theory provides a rational interpretation of the three regions in rough sets. Corresponding to the three positive, negative, and boundary regions in rough sets, this theory shows the regions of acceptance, rejection, and non-commitment in a ternary classification [17]. In this study, we use the three-way decision rule to analyze and extract useful information hidden in Table 1.

Remark 1.1

“Score of Early Semester” in Table 1 refers to the students’ calculus (1) course scores in the last semester. “Average Score of Class Quizzes” is a weighted average of the students’ scores from six class quizzes and two chapter quizzes during this semester; three or four problems in the class quiz are completed in 15 to 20 minutes. Then, the student signs in with the student number and name within the allotted time and uploads the completed quiz to the teacher. These quizzes can appropriately reflect the students’ learning situation. “Completion Ratio of Watching Videos” is the average completion ratio of students watching the teaching videos in this semester; “Completion Grade of Homework” refers to the students’ completion grade of homework in this semester. “Score of Final Exam” is the student’s final exam score in this semester.

From Table 1, which combines the queries mentioned earlier, the following question arises.

Question 1.2

Does Table 1 accurately reflect the score of the final exam derived from the “Score of Early Semester,” “Average Score of Class Exercise,” “Completion Ratio of Watching Video,” and “Grade of Homework” completion?

To discuss Question 1.2 theoretically, useful information hidden in Table 1 must be extracted. This leads us to establish an information system S = (U, C ∪ D, f) based on Table 1, where “Score of Early Semester,” “Average Score of Class Quizzes,” “Completion Ratio of Watching Videos,” and and “Grade of Completion Homework” are taken as the condition attributes, and “Score of Final Exam” is taken as the decision attribute. Thus, the authenticity of the students’ scores described in Question 1.2 can be converted to a decision rule question described in the following question.

Question 1.3

Let S = (U, C ∪ D, f) be an information system established based on Table 1. In S = (U, C ∪ D, f), is the decision attribute derived from the condition attributes? Furthermore, how can we characterize the confidence that the decision attribute is derived from the condition attributes?

Our discussion is based around Questions 1.2 and 1.3. We apply the three-way decision rules to the information system S = (U, C ∪ D, f) and use confidence to characterize the three-way decision rules, which answers Question 1.3. Based on these, we check the accuracy of whether “Score of Final Exam” is derived from “Score of Early Semester,” “Average Score of Class Quizzes,” “Completion Ratio of Watching Videos,” and “Grade of Completion Homework” in Table 1. The results of this study provide a new theoretical analysis method for online teaching and learning information for calculus, which will be helpful for teachers to develop other online courses accurately.

The basic concepts for rough-set theory and decision rule can be found in [8,13,14,16,18,19].

Notation 1.4

(1) For a finite set B, |B| denotes the cardinal of B.

(2) For a collection ℱ₁, ℱ₂, ⋯, ℱ_k of families of sets, ∧{ℱ_i : i = 1, 2, ⋯, k} denotes a family of sets:

Definition 1.5

S = (U, C ∪ D, f) is called an information system, where

(1) U, a nonempty finite set, is called the universe of discourse.

(2) C ∪ D is a finite set of attributes, where C and D are disjoint nonempty sets of condition attributes and decision attributes, respectively.

(3) f is an information function defined as U × (C ∪ D). For each u ∈ U and x ∈ C, f(u, x) is called a condition attribute value. For each u ∈ U and x ∈ D, f(u, x) is called a decision attribute value.

To apply the three-way decision rules for the theoretical analysis of online learning data of the calculus course (2), we must establish an information system S = (U, C ∪ D, f) based on Table 1.

(1) Let U = {u₁, u₂, ⋯, u₃₀} represent a set of 30 students, where u₁, u₂, ⋯, u₃₀ denote 30 students.

(2) Let C = {c₁, c₂, c₃, c₄}, where c₁, c₂, c₃, c₄ denote the “Score of Early Semester”, “Average Score of Class Quizzes”, “Completion Ratio of Watching Videos”, and “Completion Grade of Homework”, respectively.

(3) Let D = {d}, where d denotes “Score of Final Exam”.

According to the general rule of statistical grouping (e.g., refer to [20]), we divide the condition attributes into four groups. For condition attributes c₁, c₂ and c₃, the group interval

where max and min are the maximum and minimum values, respectively, of the set of group data.

(i) For the condition attribute c₁, the group interval hc1=(95+0.5)-(42-0.5)4=13.5. According to the equidistant grouping method, the open interval (41.5, 95.5) can be divided into four intervals:(41.5, 55.0), [55.0, 68.5), [68.5, 82.0), and [82.0, 95.5). Let c₁₁ indicate “Score of Early Semester” between 82.5 and 95.5, let c₂₁ indicate “Score of Early Semester” between 68.5 and 82.0, let c₃₁ indicate “Score of Early Semester” between 55.0 and 68.5, and let c₄₁ indicate “Score of Early Semester” between 41.5 and 55.0.

(ii) For the condition attribute c₂, the group interval hc2=(80+0.5)-(25-0.5)4≐14.0. According to the equidistant grouping method, the open interval (24.5, 80.5) can be divided into four intervals: (24.5, 38.5), [38.5, 52.5), [52.5, 66.5), and [66.5, 80.5). Let c₁₂ indicate “Average Score of Class Quizzes” between 66.5 and 80.5, let c₂₂ indicate “Average Score of Class Quizzes” between 52.5 and 66.5, let c₃₂ indicate “Average Score of Class Quizzes” between 38.5 and 52.5, and let c₄₂ indicate “Average Score of Class Quizzes” between 24.5 and 38.5.

(iii) For the condition attribute c₃, the group interval hc3=(99+0.5)-(21-0.5)4≐19.8. According to the equidistant grouping method, the open interval (20.5, 99.5) can be divided into four intervals: (20.5, 40.3), [40.3, 60.1), [60.1, 79.9), and [79.9, 99.5). Let c₁₃ indicate “Completion Ratio of Watching Videos” between 79.9 and 99.5, let c₂₃ indicate “Completion Ratio of Watching Videos” between 60.1 and 79.9, let c₃₃ indicate “Completion Ratio of Watching Videos” between 40.3 and 60.1, and let c₄₃ indicate “Completion Ratio of Watching Videos” between 20.5 and 40.3.

(iv) For the condition attribute c₄, Let c₁₄ indicate “Completion Grade of Homework” for A, let c₂₄ indicate “Completion Grade of Homework” for B, let c₃₄ indicate “Completion Grade of Homework” for C, and let c₄₄ indicate “Completion Grade of Homework” for D.

(v) For the decision attribute d, the group interval hd=(91+0.5)-(41-0.5)4=12.8. According to the equidistant grouping method, the open interval (40.5, 91.5) can be divided into four intervals: I_d1 = [78.9, 91.5), I_d2 = [66.1, 78.9), I_d3 = [53.3, 66.1) and I_d4 = (40.5, 53.3). Let d₁ indicate “Score of Final Exam” between 78.9 and 91.5, let d₂ indicate “Score of Final Exam” between 66.1 and 78.9, let d₃ indicate “Score of Final Exam” between 53.3 and 66.1, and let d₄ indicate “Score of Final Exam” between 40.5 and 53.3.

We now define the information function f: for each u ∈ U and d ∈ I_dj, set f(u, d) = d_j for some j = 1, 2, 3, 4. Similarly, for each u ∈ U, we define f(u, c_i) = c_ji for some i, j = 1, 2, 3, 4. Consequently, the information function f defined on U × (C ∪ D) is constructed.

From (i)-(v), an information system S = (U, C ∪ D, f) based on Table 1 is established. Furthermore, S = (U, C ∪ D, f) can be expressed as a decision table (see Table 2), whose columns are labeled by elements of C ∪ D and rows are labeled by elements of U. For each u ∈ U and x ∈ C ∪ D, f(u, x) lies in the cross of the row labeled by u and column labeled by x.

Now, we provide some simple results that are useful for our discussion.

Remark 3.1

An information system S = (U, C ∪ D, f) can be expressed as a data table, which is called a decision table, whose columns are labeled by elements of C ∪ D, rows are labeled by elements of U; f(u, x) lies in the cross of the row labeled by u and the column labeled by x.

Notation 3.2

Let S = (U, C ∪ D, f) be an information system.

(1) For x ∈ C ∪ D, we define the equivalence relation ~ on U as follows: u_i ~ u_j ⇐⇒ f(u_i, x) = f(u_j, x) for some i, j. U/x denotes the family comprising all equivalence classes with respect to ~.

(2) ∧{U/c : c ∈ C} is a partition of U denoted as U/C. The equivalence relation induced by U/C is denoted as C.

Definition 3.3

Let S = (U, C ∪ D, f) be an information system and u ∈ U. An equivalence class containing u with respect to the equivalence relation C is denoted by C(u) and is called a condition attribute granule.

Definition 3.4

Let S = (U, C ∪ D, f) be an information system and set X ⊆ U.

(1) apr(X) = {u : u ∈ U and C(u) ⊆ X} is the lower approximation of X (with respect to C).

(2) apr¯(X)={u:u∈U and C(u)∩X≠∅} is the upper approximation of X (with respect to C).

Lemma 3.5

Let S = (U, C ∪ D, f) be an information system and set X ⊆ U. Then, the following holds:

(1) apr(X) = ∪ {C(u) : u ∈ U and C(u) ⊆ X}.

(2) apr¯(X)=∪{C(u):u∈U and C(u)∩X≠∅}.

Definition 3.6

Let S = (U, C ∪ D, f) be an information system and set X ⊆ U.

(1) POS(X) = apr(X) is the positive region of X.

(2) NEG(X)=U-apr¯(X) is the negative region of X.

(3) BND(X)=apr¯(X)-apr_(X) is the boundary region of X.

Definition 3.7

Let S = (U, C ∪ D, f) be an information system, u ∈ U and X ∈ U/D.

(1) If a rule allows us to accept u as a member of X, this rule is called a positive decision rule, which is denoted as Des(C(u))→PDes(X).

(2) If a rule allows us to reject u as a member of X, this rule is called a negative decision rule, which is denoted as Des(C(u))→NDes(X).

(3) If a rule allows us to make an uncertain decision about whether we accept u as a member of X, then this rule is called a boundary decision rule, which is denoted as Des(C(u))→BDes(X).

Here, the positive, negative, and boundary decision rules are called the three-way decision rules.

In an information system S = (U, C ∪ D, f), the positive, negative, and boundary regions of decision class X ∈ U/D can be constructed.

Lemma 3.8

Let S = (U, C ∪ D, f) be an information system, u ∈ U and X ∈ U/D. Then, the following holds:

(1) If C(u) ⊆ POS(X), then Des(C(u))→PDes(X).

(2) If C(u) ⊆ NEG(X), then Des(C(u))→NDes(X).

(3) If C(u) ⊆ BND(X), Then Des(C(u))→BDes(X).

We provide an easier method to derive the positive, negative, and boundary decision rules.

Lemma 3.9

Let S = (U, C ∪ D, f) be an information system, u ∈ U and X ∈ U/D. Then, the following holds:

(1) u ∈ POS(X) if and only if C(u) ⊆ POS(X);

(2) u ∈ NEG(X) if and only if C(u) ⊆ NEG(X);

(3) u ∈ BND(X) if and only if C(u) ⊆ BND(X);

The following theorem can be obtained from Lemmas 3.8 and 3.9.

Theorem 3.10

Let S = (U, C ∪ D, f) be an information system, u ∈ U and X ∈ U/D. Then, the following holds:

(1) If u ∈ POS(X), then Des(C(u))→PDes(X).

(2) If u ∈ BND(X), Des(C(u))→BDes(X).

(3) If u ∈ NEG(X), then Des(C(u))→NDes(X).

The confidence of three-way decision rules on an information system was introduced by Yao and his colleague [12,15].

Definition 3.11

Let S = (U, C ∪ D, f) be an information system, u ∈ U, X ∈ U/D and Λ ∈ {P, B, N}. Put

Then, conf(Des(C(u))→ΛDes(X)) is the confidence of the decision rule Des(C(u))→ΛDes(X).

We have the following proposition:

Proposition 3.12

Let S = (U, C ∪ D, f) be an information system, u ∈ U, Λ ∈ {P, B, N} and X ∈ U/D. Then, the following holds:

(1) conf(Des(C(u))→ΛDes(X))=1 if and only if Des(C(u))→PDes(X);

(2) conf(Des(C(u))→ΛDes(X))=0 if and only if Des(C(u))→NDes(X).

(3) 0<conf(Des(C(u))→ΛDes(X))<1 if and only if Des(C(u))→BDes(X).

In the following sections, the information system S = (U, C ∪ D, f) is expressed in Table 2.

In this section, we discuss three-way decision rules for S = (U, C ∪ D, f). First, we provide some related partitions of U obtained from Table 2.

Proposition 4.1

The following partitions of U hold for S = (U, C ∪ D, f).

(1) U/C = {{u₃}, {u₄}, {u₆}, {u₇}, {u₁₁}, {u₁₃}, {u₁₄}, {u₁₆}, {u₁₇}, {u₁₈, {u₁₉}, {u₂₀}, {u₂₁}, {u₂₃}, {u₁, u₂₄}, {u₂, u₉}, {u₁₂, u₁₅}, {u₅, u₂₅, u₂₉}, {u₈, u₁₀, u₂₈}, {u₂₂, u₂₆, u₂₇, u₃₀}}.

(2) U/d = {{u₂, u₃, u₅, u₇, u₁₀, u₁₃, u₁₈, u₂₂, u₂₅, u₂₆, u₂₇, u₂₈, u₂₉, u₃₀}, {u₁, u₄, u₈, u₉, u₁₁, u₁₂, u₁₄, u₁₇, u₂₃}, {u₆, u₁₅, u₁₆, u₂₄}, {u₁₉, u₂₀, u₂₁}}.

From Proposition 4.1, we get some conditional attribute granules in S = (U, C ∪ D, f).

Proposition 4.2

The following conditional attribute granules hold for S = (U, C ∪ D, f).

(1) C(u) = {u} for each u ∈ {u₃, u₄, u₆, u₇, u₁₁, u₁₃, u₁₄, u₁₆, u₁₇, u₁₈, u₁₉, u₂₀, u₂₁, u₂₃}.

(2) C(u) = X for each u ∈ X and X ∈ {{u₁, u₂₄}, {u₂, u₉}, {u₁₂, u₁₅}, {u₅, u₂₅, u₂₉}, {u₈, u₁₀, u₂₈}, {u₂₂, u₂₆, u₂₇, u₃₀}}.

Let U/d = {D₁, D₂, D₃, D₄}, where D₁ = {u₂, u₃, u₅, u₇, u₁₀, u₁₃, u₁₈, u₂₂, u₂₅, u₂₆, u₂₇, u₂₈, u₂₉, u₃₀}, D₂ = {u₁, u₄, u₈, u₉, u₁₁, u₁₂, u₁₄, u₁₇, u₂₃}, D₃ = {u₆, u₁₅, u₁₆, u₂₄} and D₄ = {u₁₉, u₂₀, u₂₁}. For u ∈ D_i, the value of the decision attribute is d_i (for some i = 1, 2, 3, 4). By Lemma 3.5, Proposition 4.1, and Proposition 4.2, we have the upper and lower approximations of D₁, D₂, D₃, D₄.

Proposition 4.3

The following upper and lower approximations of D₁, D₂, D₃, D₄ hold for S = (U, C ∪ D, f).

(1) apr(D₁) = {u₃, u₅, u₇, u₁₃, u₁₈, u₂₂, u₂₅, u₂₆, u₂₇, u₂₉, u₃₀}.

apr¯(D1)={u2,u3,u5,u7,u8,u9,u10,u13,u18,u22,u25,u26,u27,u28,u29,u30}.

(2) apr(D₂) = {u₄, u₉, u₁₁, u₁₄, u₁₇, u₂₃}.

apr¯(D2)={u1,u4,u8,u9,u10,u11,u12,u14,u15,u17,u23,u24,u28}.

(3) apr(D₃) = {u₆, u₁₆},

apr¯(D3)={u1,u6,u12,u15,u16,u24}.

(4) apr(D₄) = {u₁₉, u₂₀, u₂₁}.

apr¯(D4)={u19,u20,u21}.

Now, by Definition 3.6 and Proposition 4.3, we discuss the three-way decision rules for S = (U, C ∪ D, f). We have the following positive, negative, and boundary regions of D₁, D₂, D₃, D₄.

Theorem 4.4

The following positive, negative, and boundary regions of D₁, D₂, D₃, D₄ hold for S = (U, C ∪ D, f).

(1) POS(D₁) = {u₃, u₅, u₇, u₁₃, u₁₈, u₂₂, u₂₅, u₂₆, u₂₇, u₂₉, u₃₀}.

NEG(D₁) = {u₁, u₄, u₆, u₁₁, u₁₂, u₁₄, u₁₅, u₁₆, u₁₇, u₁₉, u₂₀, u₂₁, u₂₃, u₂₄}.

BND(D₁) = {u₂, u₈, u₉, u₁₀, u₂₈}.

(2) POS(D₂) = {u₄, u₉, u₁₁, u₁₄, u₁₇, u₂₃}.

NEG(D₂) = {u₂, u₃, u₅, u₆, u₇, u₁₃, u₁₆, u₁₈, u₁₉, u₂₀, u₂₁, u₂₂, u₂₅, u₂₆, u₂₇, u₂₉, u₃₀}.

BND(D₂) = {u₁, u₈, u₁₀, u₁₂, u₁₅, u₂₄, u₂₈}.

(3) POS(D₃) = {u₆, u₁₆}.

NEG(D₃) = {u₂, u₃, u₄, u₅, u₇, u₈, u₉, u₁₀, u₁₁, u₁₃, u₁₄, u₁₇, u₁₈, u₁₉, u₂₀, u₂₁, u₂₂, u₂₃, u₂₅, u₂₆, u₂₇, u₂₈, u₂₉, u₃₀}.

BND(D₃) = {u₁, u₁₂, u₁₅, u₂₄}.

(3) POS(D₄) = {u₁₉, u₂₀, u₂₁}.

NEG(D₄) = {u₁, u₂, u₃, u₄, u₅, u₆, u₇, u₈, u₉, u₁₀, u₁₁, u₁₂, u₁₃, u₁₄, u₁₅, u₁₆, u₁₇, u₁₈, u₂₂, u₂₃, u₂₄, u₂₅, u₂₆, u₂₇, u₂₈,

u₂₉, u₃₀}.

BND(D₄) = ∅︀.

By Theorems 3.10 and 4.4, we can easily obtain the three-way decision rules for S = (U, C ∪ D, f). Table 3 lists Des(C(u))→ΛDes(Di) for all u ∈ U and i ∈ {1, 2, 3, 4}, where Λ ∈ {P, N, B}. Here, the decision rule Des(C(u))→ΛDes(Di) is simply denoted by C(u)→ΛDi for u ∈ U, Λ ∈ {P, N, B} and all i ∈ {1, 2, 3, 4} (This notation is shown in Table 4).

The three-way decision rules for S = (U, C ∪ D, f) can also be shown by its confidence. In other words, we can use conf(Des(C(u))→ΛDes(Di)) to characterize Des(C(u))→ΛDes(Di), where u ∈ U, Λ ∈ {P, N, B} and all i ∈ {1, 2, 3, 4}. For example, for u₁, C(u₁) = {u₁, u₂₄}, C(u₁) ∩ D₁ = ∅︀, C(u₁) ∩ D₂ = {u₁}, C(u₁) ∩ D₃ = {u₂₄} and C(u₁) ∩ D₄ = ∅︀. By Definition 3.11,

Thus, we have confidence in the three-way decision rules on S = (U, C ∪ D, f)(see Table 4).

Tables 3 and 4 give the three-way decision rules on S = (U, C ∪ D, f) and the confidence of the three-way decision rules, respectively, which answers Question 1.3 (see Conclusion 5.2 for more detailed explanations).

In this section, we focus on Question 1.2. First, we provide a remark on the three-way decision rules on S = (U, C ∪ D, f).

Remark 5.1

The following are some semantic interpretations for the three-way decision rules on S = (U, C ∪ D, f) in the sense of [15], where i ∈ {1, 2, 3, 4}. By Proposition 3.12,

(1) For u ∈ U, if Des(C(u))→PDes(Di) (equivalently, conf(Des(C(u))→PDes(Di))=1), then the decision attribute value derived from condition attribute values is d_i.

(2) For u ∈ U, if Des(C(u))→NDes(Di) (equivalently, conf(Des(C(u))→ΛDes(Di))=0), then the decision attribute value derived from the condition attribute values is not d_i.

(3) For u ∈ U, if Des(C(u))→BDes(Di) (equivalently, 0<conf(Des(C(u))→ΛDes(Di))<1), then it is uncertain whether the decision attribute value derived from the condition attribute values is d_i.

From Table 3 (or Table 4) and Remark 5.1, we have the following more detailed explanations of the three-way decision rules on S = (U, C ∪ D, f).

Conclusion 5.2

The following are true for S = (U, C ∪ D, f).

(1) For u = u₁, the decision attribute value derived from the condition attribute values is not d_i because Des(C(u))→NDes(Di), where i = 1, 4. However, it is uncertain whether the decision attribute value derived from the condition attribute values is d_j because Des(C(u))→BDes(Dj), where j = 2, 3.

(2) For u = u₂, the decision attribute value derived from the condition attribute values is not d_i because Des(C(u))→NDes(Di), where i = 3, 4. However, it is uncertain whether the decision attribute value derived from condition attribute values is d_j because Des(C(u))→BDes(Dj), where j = 1, 2.

(3) For u = u₃, the decision attribute value derived from the condition attribute values is d₁ because Des(C(u))→PDes(D1).

(4) For u = u₄, the decision attribute value derived from the condition attribute values is d₂ because Des(C(u))→PDes(D2).

(5) For u = u₅, the decision attribute value derived from the condition attribute values is d₁ because Des(C(u))→PDes(D1).

(6) For u = u₆, the decision attribute value derived from the condition attribute values is d₃ because Des(C(u))→PDes(D3).

(7) For u = u₇, the decision attribute value derived from the condition attribute values is d₁ because Des(C(u))→PDes(D1).

(8) For u = u₈, the decision attribute value derived from the condition attribute values is not d_i because Des(C(u))→NDes(Di), where i = 3, 4. However, it is uncertain whether the decision attribute value derived from condition attribute values is d_j because Des(C(u))→BDes(Dj), where j = 1, 2.

(9) For u = u₉, the decision attribute value derived from the condition attribute values is not d_i because Des(C(u))→NDes(Di), where i = 3, 4. However, it is uncertain whether the decision attribute value derived from condition attribute values is d_j because Des(C(u))→BDes(Dj), where j = 1, 2.

(10) For u = u₁₀, the decision attribute value that is derived from the condition attribute values is not d_i because Des(C(u))→NDes(Di), where i = 3, 4. However, it is uncertain whether the decision attribute value that is derived from condition attribute values is d_j because Des(C(u))→BDes(Dj), where j = 1, 2.

(11) For u = u₁₁, the decision attribute value derived from the conditional attribute values is d₂ because Des(C(u))→PDes(D2).

(12) For u = u₁₂, the decision attribute value derived from the condition attribute values is not d_i because Des(C(u))→NDes(Di), where i = 1, 4. However, it is uncertain whether the decision attribute value derived from the condition attribute values is d_j because Des(C(u))→BDes(Dj), where j = 2, 3.

(13) For u = u₁₃, the decision attribute value derived from the condition attribute values is d₁ because Des(C(u))→PDes(D1).

(14) For u = u₁₄, the decision attribute value derived from the condition attribute values is d₂ because Des(C(u))→PDes(D2).

(15) For u = u₁₅, the decision attribute value derived from the condition attribute values is not d_i because Des(C(u))→NDes(Di), where i = 1, 4. However, it is uncertain whether the decision attribute value derived from the condition attribute values is d_j because Des(C(u))→BDes(Dj), where j = 2, 3.

(16) For u = u₁₆, the decision attribute value derived from the conditional attribute values is d₃ because Des(C(u))→PDes(D3).

(17) For u = u₁₇, the decision attribute value derived from the conditional attribute values is d₂ because Des(C(u))→PDes(D2).

(18) For u = u₁₈, the decision attribute value derived from the conditional attribute values is d₁ because Des(C(u))→PDes(D1).

(19) For u = u₁₉, the decision attribute value derived from the condition attribute values is d₄ because Des(C(u))→PDes(D4).

(20) For u = u₂₀, the decision attribute value derived from the condition attribute values is d₄ because Des(C(u))→PDes(D4).

(21) For u = u₂₁, the decision attribute value derived from the condition attribute values is d₄ because Des(C(u))→PDes(D4).

(22) For u = u₂₂, the decision attribute value derived from the conditional attribute values is d₁ because Des(C(u))→PDes(D1).

(23) For u = u₂₃, the decision attribute value derived from the conditional attribute values is d₁ because Des(C(u))→PDes(D1).

(24) For u = u₂₄, the decision attribute value derived from the condition attribute values is not d_i because Des(C(u))→NDes(Di) where i = 2, 4. However, it is uncertain whether the decision attribute value derived from the condition attribute values is d_j because Des(C(u))→BDes(Dj), where j = 1, 3.

(25) For u = u₂₅, the decision attribute value derived from the conditional attribute values is d₁ because Des(C(u))→PDes(D1).

(26) For u = u₂₆, the decision attribute value derived from the conditional attribute values is d₁ because Des(C(u))→PDes(D1).

(27) For u = u₂₇, the decision attribute value derived from the conditional attribute values is d₁ because Des(C(u))→PDes(D1).

(28) For u = u₂₈, the decision attribute value that is derived from the condition attribute values is not d_i because Des(C(u))→NDes(Di), where i = 3, 4. However, it is uncertain whether the decision attribute value derived from the conditional attribute values is d_j because Des(C(u))→BDes(Dj), where j = 1, 2.

(29) For u = u₂₉, the decision attribute value derived from the conditional attribute values is d₁ because Des(C(u))→PDes(D1).

(30) For u = u₃₀, the decision attribute value derived from the condition attribute values is d₁ because Des(C(u))→PDes(D1). In S = (U, C ∪ D, f), Conclusion 5.2 gives a theoretical answer to Question 1.3.

Conclusion 5.3

For online teaching data of calculus (2), the final scores of most students are credible.

For students: 2019488007, 2019488008, 2019488009, 2019 488011, 2019488013, 2019488022, 2019488027, 2019488028, 2019488032 2019488034, 2019488036, 2019488038, 2019488 039, 2019488040, 2019488041, 2019488045, 2019488049, 201 9488050, 2019488054, 2019488057 and 2019488060, their scores are credible. For students: 2019488002, 2019488005 2019488014, 2019488015, 2019488020, 2019488025, 201948 8030, 2019488046 and 2019488055, their scores are uncertain.

Thus, the scores of the online exam are mostly credible. This provides a theoretical basis for online teaching and examinations.

Although this paper focuses on the theoretical analysis of online learning data of calculus (2), it still has practical significance for other online courses. However, a more practical analysis is required to determine whether this theory is suitable for investigating other problems.

Table 1. Online learning data of calculus (2) course.

Table 2. Decision table.

Table 3. Three-way decision rules on S = (U, C ∪ D, f).

Table 4. Confidence of the three-way decision rules on S = (U, C ∪ D, f).