Exploration of the Concept of Relation and Function in the Quran with the Theme of Q.S. Ar-Rahman

ABSTRACT


INTRODUCTION
Education is a powerful center of excellence in preparing the quality and formation of human character to face global challenges [1].Its role in producing highquality human resources and good character that can contribute to the nation and state is undeniable [2].National education is a clear testament to this, aiming to cultivate individuals who are faithful and devoted to God Almighty, noble, healthy, knowledgeable, capable, creative, independent, and able to become democratic and responsible citizens [3].
To achieve this goal, the cultivation of character and morals to form a person of faith and piety is not only given in certain subjects [4], [5].Still, it is given integratively in every subject, including mathematics [6]- [8].In other words, Integrated mathematics not only plays a role in training students in problemsolving [9], [10], and thinking skills [11], [12] but also has a responsibility for the moral and behavioral aspects of students.[13], [14].Therefore, integrated learning of mathematics is essential in education.[15].Some integration of mathematics has been conducted across different fields.For example, applying mathematical concepts to culture and customs is commonly referred to as ethnomathematics [16]- [18].Whereas mathematics integrated with Islamic values to develop a person with Islamic understanding and behavior is termed Islamic-integrated mathematics [19]- [21].
Furthermore, there is a growing demand for Islamic-integrated mathematics learning, as evidenced by the numerous models implemented in primary, secondary, and tertiary education institutions.[21], [22].A mathematics olympiad contains integrated Islamic mathematics, known as Kompetisi Sains Madrasah (KSM) or the Madrasah Science Competition [23].In addition, Islamic integrated mathematics has also been applied in various modules, learning media, and other learning tools [24]- [27].
The integration of mathematics with Islam can be done through various sources and guidelines derived from Islam, such as the Quran, hadiths, Islamic history, and others [28]- [31].As the holy book for Muslims, the Quran is the most outstanding guide to life, grace, and guidance from Allah SWT for human beings [20], [32].It includes hudan li al-nas (guidance for humankind), albayyinat (explanation), al-furqan (the criterion), and syifa'un wa Rahmah (compassionate medicine) [33].The Quran includes science teachings, particularly in biology, astronomy, geology, and mathematics, and stories, laws, and Muslim guidelines.[34], [35].
However, previous studies have only discussed simple mathematical concepts derived from the Quran, such as sets, numbers, and their operations.The mathematical concepts that students learn from elementary school to college level are diverse, one of which is the concept of relations and functions [26], [48], [49].Therefore, the researcher needed to explore the concept of relation and function from the Quran.
To simplify and deepen the mathematical study, the researcher restricts the letters in the Quran that are studied.This study uses the verses in Surah Ar-Rahman as the object to be explored mathematically.Surah Ar-Rahman was chosen because the verses in the surah are relatively short, but there are many of them, so it will be easier to calculate the letters.In addition, there are verses in Surah Ar-Rahman repeated 31 times in a certain range of verses, giving the impression of a form of mathematical pattern and uniqueness in the letter.Therefore, this research explores the concept of relation and function in the Quran with the theme of Q.S. Ar-Rahman.

‫ب‬ ‫ف‬ ُ "So which of the favors of your lord would you deny?" (Q.S. Ar-Rahman verse 13)
The verse is repeated in Q.S Ar-Rahman 31 times.It starts in the 13th verse and ends in the 77th verse.The entire repetition of the verse is presented in Table 1.

Numerical Value (Gematria) of Hijaiyah Letters
The numerical value in Hijaiyah letters in Indonesia is Abajadun [37].In common parlance, this study is called Gematria, which examines the numerical values of an alphabet and relates them to certain phenomena [50], [51].The numerical values of the hijaiyyah letters are shown in Table 2.

METHODS
This research uses a qualitative approach with the type of literature study.The literature study was conducted through the study of the Quran, mathematics books related to relations and functions, proceedings articles, and journal articles.Then, the researchers grouped the literature according to the research theme.Furthermore, researchers conducted an in-depth and detailed review of each literature.
The data collection technique uses secondary sources.That means the research data is not provided directly to researchers but through intermediaries such as books and articles.This research uses four stages of activities, namely (1) preparing the equipment needed, (2) preparing a working bibliography, (3) conducting a review, and (4) recording and analyzing research materials [52].
The results of the analysis were then reduced and synthesized into specific data.Furthermore, the analysis results are written and compared with other studies in the discussion section.That is so researchers can obtain clear conclusions to answer research questions [53].

The Concept of Relation in Q.S Ar-Rahman Verses 1-5
As for Q.S Ar-Rahman verses 1-5, there are various letter arrangements with many letters in each verse, as shown in Table 3 below.

Concept of Relation from Composing Letters to Verse Number
In mathematics, a relation is a set of ordered pairs.A relation is also defined as a relationship between a domain and a codomain [49].Each relationship in the domain and codomain of the relation has no restrictions, so relations have differences with functions [54].

Relation from Number of Letters to Verse Number
Suppose set  is the number of letters consisting of 6,9,10,17, which can be written as  = {6,9,10,17}.The set  is the verse number consisting of 1,2,3,4,5, which can be written as  = {1,2,3,4,5}.Thus, the relation of set A to set B can be presented in Figure 2 below.Meanwhile, the relation from many letters to verse number can be presented with sequential pairs as follows.

Function Concept in Q.S. Ar-Rahman
A function is a relationship between a domain and a codomain, where every domain member is exactly paired with one member of the codomain [49].That means that functions are a specialized form of relation.Every function is undoubtedly a relation, but every relation is not necessarily a function.[54].
In Q.S Ar-Rahman verse 1, functions are formed, namely bijective functions and surjective functions.Bijective functions are shown from the arrangement of letters and numerical values.The surjective function is compared to numerical values and value categories.The arrangement of letters, numerical values, and numerical value categories are shown in Table 4

Bijective Function
A bijective function is a function whose codomain members cannot be unpaired, and every member of the codomain must pair with exactly one member of the domain [49].A bijective function is a function whose codomain members cannot be unpaired, and every member of the codomain must pair with exactly one member of the domain.[26].
Based on Q.S. Ar-Rahman verse 1 and Table 4, the known arrangement of letters and numerical values forms a bijective function.Suppose set A is a sequence of letters in Q.S. Ar-Rahman verse 1, and set B is the numerical Hijaiyah letters.So, it can be written  = ‫,ا(‬ ‫,ل‬ ‫ر‬ , ‫ح‬ , ‫م‬ , ‫)ن‬ and  = (1,30,200,8,40,50).The bijective function from set  to set  is presented in Figure 5.

Surjective Function
A surjective function is a function whose codomain members cannot be unpaired.In a surjective function, each member of the codomain can be paired with more than one member of the domain [49].
In Q.S Ar-Rahman verse 1, numerical values and value categories form a surjective function.Suppose set A means numerical value in Q.S Ar-Rahman verse 1, and set B means value category.So it can be written as  = (1,30,200,8,40,50), and  = (, ).The surjective function of set A to set B is presented in Figure 6.

Injektive Function
An injective function is one whose codomain members can only pair with one domain member.In an injective function, the members of the codomain may not be paired.The injective function in Surah Ar-Rahman is obtained from the arrangement and number of letters in Q.S. Ar-Rahman verse 13.That is shown in Table 5.

Presentation of Function Concept a. Function Graph y = f(x)
Besides using arrow diagrams, functions can also be represented with function graphs.The function graph of a bijective function is shown in Figure 8, a surjective function in Figure 9, and an injective function in Figure 10.The representation of the set of consecutive pairs on the surjective function, namely f = {(1, odd), (8, even), (30, even), (40, even), (50, even), (200, even).While the representation of the set of ordered pairs on the injective function, namely f = ‫,ف({‬ 1), ( ‫ب‬ , 3), ‫,ا(‬ 4), ‫,ي(‬ 1), ‫,ل(‬ 1), ‫,ء(‬ 1), ‫,ر(‬ 1), ‫,ك(‬ 2), ‫,م(‬ 1), ‫,ت(‬ 1), ‫,ذ(‬ 1), ‫,ن(‬ 1)}.mathematical exploration of Q.S Ar-Rahman.The concept of relation is shown in the number of letters to the verse number and the number of letters to the verse number in Q.S Ar-Rahman verses 1 to 5. The relations can be presented with arrow diagrams, cartesian diagrams, and sets of consecutive pairs.The function concepts found are bijective, injective, and surjective functions.The concept of the bijective function is found in the arrangement of letters to the numerical value of Q.S Ar-Rahman: 1.The concept of the surjective function is found in the numerical value to the numerical value category in Q.S Ar-Rahman verse 1.At the same time, the injective function is found in the arrangement of letters to many letters in Q.S Ar-Rahman verse 13.The presentation of functions can be done with arrow diagrams, function graphs, and sequential pairs.
It is hoped that the mathematics of Islamic integration on the concept of relations and functions can provide a new color for the formation of students' problem-solving skills while increasing students' faith when learning mathematics at school.

Figure 1 .
Figure 1.The relation of verse letter arrangement to verse number

Figure 2 . 4 . 1 . 3
Figure 2. The relation from the Number of Letters to Verse Number Based on Figure 2, the relationship between set  and set  forms a relation.That means the relation has a domain, codomain, and range.The domain part is shown in set

Figure 3 .
Figure 3. Presentation of Relation from Letter Arrangement to Verse Number with Cartesian Diagram

Figure 5 .
Figure 5. Bijective Function in Q.S. Ar-Rahman verse 1 Based on Figure 5, the relationship between set  and set  forms a bijective function.The domain formed is  = ‫,ا{‬ ‫,ل‬ ‫ر‬ , ‫ح‬ , ‫م‬ , ‫.}ن‬At the same time, the codomain and range are indicated by the set of  = {1, 30, 200, 8, 40, 50}.4.2.2 Surjective FunctionA surjective function is a function whose codomain members cannot be unpaired.In a surjective function, each member of the codomain can be paired with more than one member of the domain[49].

Figure 6 .
Figure 6.Surjective Function in Q.S. Ar-Rahman verse 1 Based on Figure 6, the relationship between set  and set  forms a surjective function.The domain formed is the set of  = {1, 30, 200, 8, 40, 50}.In contrast, the codomain and range formed, shown by the set of  = {, }.4.2.3 Injektive FunctionAn injective function is one whose codomain members can only pair with one domain member.In an injective function, the members of the codomain may not be paired.The injective function in Surah Ar-Rahman is obtained from the arrangement and number of letters in Q.S. Ar-Rahman verse 13.That is shown in Table5.Table 5. Letter Arrangement and Number of Letters in Q.S. Ar-Rahman verse 13

Table 1 .
Repetition of verses in Q.S Ar-Rahman

Table 2 .
Numerical Value of Hijaiyah Letters

Table 3 .
Arrangement and Number of Letters of Q.S. Ar-Rahman Verses 1-5

Table 5 .
Letter Arrangement and Number of Letters in Q.S. Ar-Rahman verse 13