INTRODUCTION TO COMPUTER SCIENCE HANDOUT #3. THE RELATIONAL DATA MODEL K5 & K6, Computer Science Department, Văn Lang University Second semester Feb, 2002 Instructor: Trần Đức Quang Major themes: 1. Binary Relations 2. Relations 3. Relational Algebra Reading: Sections 7.7, 7.10, 8.2, and 8.7. 3.1 BINARY RELATIONS Consider two sets of students: MS = {Tiến, Hùng, Khoa, Việt}, and FS = {Lan, Cúc, Huệ}. Suppose that we have two pairs of lovers: Tiến and Cúc, Việt and Lan. Others may love someone else not in the sets. We say that the binary relation love on the sets MS and FS has two tuples: (Tiến, Cúc) and (Việt, Lan). The other tuples, say (Khoa, Huệ) or (Hùng, Lan), are not in love. We can now define binary relations formally. First, let A and B be two sets. The product or Cartesian product of A and B, denoted A × B, is defined as the set of pairs in which the first component is chosen from A and the second component is chosen from B. That is, A × B = { (a, b) | a ∈ A and b ∈ B } A binary relation R on A and B is a subset of the product A × B. In the previous example, we have love = { (Tiến, Cúc), (Việt, Lan) } and may write in a predicate form: love(Tiến, Cúc) = true love(Việt, Lan) = true love(Hùng, Huệ) = false 18 INTRODUCTION TO COMPUTER SCIENCE: HANDOUT #3. THE RELATIONAL DATA MODEL The last must be false since it is not in the relation. We can use an infix notation for binary relations, so that a binary relation like < can be written between the componets of pairs in the relation. For example, we may write 1<2 instead of "(1,2)∈<" or "<(1,2)=true". Some special properties Binary relations may have some useful properties: transitivity, reflexivity, symmetry, and antisymmetry. Reflexivity A relation R on a set A is called reflexive if (a, a) ∈ R for every element a ∈ A. Symmetry A relation R on a set A is called symmetric if (b, a) ∈ R whenever (a, b) ∈ R for a, b ∈ A. Antisymmetry A relation R on a set A such that (a, b) ∈ R and (b, a) ∈ R only if a = b, for a, b ∈ A, is called antisymmetric. Transitivity A relation R on a set A is called transitive if whenever (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R, for a, b, c ∈ A. See more about binary relations in the textbook (partial orders, equivalence relations, equivalence classes, closures of relations). 3.2 RELATIONS In this section, we shall extend binary relations into relations of arbitrary arity. In the relational model, relations are viewed as tables; that is, we represent information in a table whose columns are given names, called attributes, and whose rows are called tuples and represent basic facts. Thus, a table has two aspects: 1. The set of column names, and 2. The rows containing the information. The set of column names (attributes) is called the scheme of the relation. For exam- ple, the table on the next page has the scheme (Course,StudentId,Grade). If we give it a name, say CSG, we may write CSG(Course,StudentId,Grade). 3.3 RELATIONAL ALGEBRA 19 The table CSG with three attributes. A collection of relations (tables) is called a database. In an enterprise, databases could contain all of the vital information for its operations. Design of a dabase and applications that access data in the database is a big problem and is beyond the scope of this course. A set of schemes of various relations in a database is called the database scheme. Notice the difference between the scheme for the database, which tells us something about how information is organized in the database, and the set of tuples in each rela- tion, which is the actual information stored in the database. 3.3 RELATIONAL ALGEBRA We introduce some basic operations on relations by examples. You can see more about those operations and others in many database textbooks. The Selection Operation If we want the tuples from the table CSG that have the Course component "CS101", we can perform a selection on this table and write: σ Course=“CS101” (CSG) where σ is the selection operator; Course="CS101" is a boolean expression that can consist of the logical operators such as AND, OR, and NOT. The result of this operation is as follows: Course StudentId Grade CS101 CS101 EE200 EE200 CS101 PH100 12345 67890 12345 22222 33333 67890 A B C B+ A− C+ Course StudentId Grade CS101 CS101 CS101 12345 67890 33333 A B A− 20 INTRODUCTION TO COMPUTER SCIENCE: HANDOUT #3. THE RELATIONAL DATA MODEL The Projection Operation Another important operation is the projection. Suppose we want the identifiers of the students in the table CSG. That is, we must eliminate all the columns but StudentId. This can be done using the projection operator, represented by the symbol π. π StudentId (CSG) and we can get an one-column table: Note that this table has only four tuples, not six as in the original one. The reason is that relations are sets, and the 1-tuples "12345" and "67890" are duplicates, thus they are the same elements and need not to be represented twice. The Join Operation Unlike the previous operations that are unary ones, the join operation is a binary oper- ation; that is, it has two operands. The join of R and S, written R S, is formed by taking each tuple r from R and each tuple s from S and comparing them. If the component of r for A i equals the com- ponent of s for B j , then we form one tuple from r and s; otherwise, no tuple is created from the pairing of r and s. We form a tuple from r and s by taking the components of r and following them by all the components of s, but omitting the component for B j , which is the same as the A i component of r anyway. Suppose we have two tables CDH and CR as follows: The table CDH. StudentId 12345 67890 22222 33333 Course Day Hour CS101 CS101 CS101 EE200 EE200 EE200 M W F Tu W Th 9AM 9AM 9AM 10AM 1PM 10AM >< A i =B j 3.4 GLOSSARY 21 The table CR. and we want to know at what times each room is occupied by some course taking a join on the Course components. The expression defining the resulting relation is: CR CDH and the value of the relation produced by this expression is as shown in the following table: We usually perform a kind of joins that equates the attributes with the same names. Such a join is called a natural join and simply written . The join in the example is a natural join and can be rewritten CH CDH. 3.4 GLOSSARY Relation: Quan hệ. See definition in text. Arity: Ngôi (của quan hệ). Binary relation: Quan hệ hai ngôi. See definition in text. n-ary relation: Quan hệ n-ngôi. Tuple: Bộ (dữ liệu). A sorted list of values, each corresponding to one component in the relation scheme. Also called a fact. Attribute: Thuộc tính. A component of a relation that is given a name. Cartesian Product: Tích, tích Descartes. Also called cross product (tích chéo). Course Room CS101 EE200 PH100 Turing Aud. 25 Ohm Hall Newton Lab. Course Room Day Hour CS101 CS101 CS101 EE200 EE200 EE200 Turing Aud. Turing Aud. Turing Aud. 25 Ohm Hall 25 Ohm Hall 25 Ohm Hall M W F Tu W Th 9AM 9AM 9AM 10AM 1PM 10AM >< Course=Course >< >< 22 INTRODUCTION TO COMPUTER SCIENCE: HANDOUT #3. THE RELATIONAL DATA MODEL Predicate: Vò từ. See Chapter 14 in the textbook for Predicate Logic. Infix notation: Ký pháp trung vò. An expression notation in which operators are betweet their operands. Related terms: prefix and postfix notation (tiền vò và hậu vò). Reflexivity: Tính phản thân. Symmetry: Tính đối xứng. Antisymmetry: Tính phản xứng. Transitivity: Tính bắc cầu. Partial order: Thứ tự bộ phận. Equivalence relation: Quan hệ tương đương. Equivalence class: Lớp tương đương. Closure: Bao đóng. Scheme: Lược đồ. Database: Cơ sở dữ liệu. Relational algebra: Đại số quan hệ. Selection: Phép chọn. Projection: Phép chiếu. Join: Phép nối. Natural Join: Nối tự nhiên. . Grade CS101 CS101 EE200 EE200 CS101 PH100 1 234 5 67890 1 234 5 22222 33 333 67890 A B C B+ A− C+ Course StudentId Grade CS101 CS101 CS101 1 234 5 67890 33 333 A B A− 20 INTRODUCTION TO COMPUTER SCIENCE: HANDOUT #3. THE RELATIONAL. INTRODUCTION TO COMPUTER SCIENCE HANDOUT #3. THE RELATIONAL DATA MODEL K5 & K6, Computer Science Department, Văn Lang University Second semester Feb, 2002 Instructor: Trần Đức. CDH. StudentId 1 234 5 67890 22222 33 333 Course Day Hour CS101 CS101 CS101 EE200 EE200 EE200 M W F Tu W Th 9AM 9AM 9AM 10AM 1PM 10AM >< A i =B j 3. 4 GLOSSARY 21 The table CR. and we want to know