Let A1,A2,...,An be attribute names with associated domains D1, D2, . . . , Dn, then R(A1: D1,A2: D2,...,An: Dn) is a relation schema. For example, Student(StudId: integer, StudName: string, Major: string) • A relation schema specifies the name and the structure of the relation. • A collection of relation schemas is called a relational database schema. * 1. List the names of all students who have borrowed a book and who are CS majors. πStName(σSTUDENTS.StId=borrows.StId (σMajor=’CS’(STUDENTS)×borrows)) * 2. List the title of books written by the author ’Silberschatz’. πTitle(σAName=’Silberschatz’ (σhas-written.DocId=BOOKS.DocID(has-written×BOOKS))) or πTitle(σhas-written.DocId=BOOKS.DocID (σAName=’Silberschatz’(has-written)×BOOKS)) * 3. As 2., but not books that have the keyword ’database’. * . . . as for 2. . . . − πTitle(σdescribes.DocId=BOOKS.DocId (σKeyword=’database’(describes)×BOOKS)) * 4. Find the name of the youngest student. πStName(STUDENTS)− πS1.StName(σS1.Age>S2.Age(ρS1(STUDENTS)×ρS2(STUDENTS))) * 5. Find the title of the oldest book. πTitle(BOOKS)−πB1.Title(σB1.Year>B2.Year(ρB1(BOOKS)×ρB2(BOOKS))) * Relational algebra is a language that is being used to explain basic relational operations and principles. * It is procedural language which instructs the system to perform a specific sequence of tasks on the database. * Relational algebra is a procedural query language with five fundamental operations. These operations include project, select, Cartesian product, union and set difference. * Projection is a unary operation denoted by the Greek letter pi (Π). Πcomma separated list of attributes(relation) * If the projection produces two identical rows, the duplicate rows must be removed since the relation is a set and it is not allowed to contain identical tuples.

* Selection is a unary operation that selects tuples that satisfy a given predicate. σselection condition(relation)
The selection condition or selection criterion can be any legally formed expression that involves :
1. Constants (i.e., members of any attribute domain)
2. Attribute names
3. Arithmetic comparisons (=, 6=, <, ≤, >, ≥)
4. Logical operators (and, or, not)

* Example σlast name=teacher name(student) results in all students who have the same name as their teacher.

* The Cartesian product is the first binary operation and it is denoted by a cross (×). The Cartesian product of two relations, r1 and r2, is written in infix notation as r1 × r2. * The binary operation union is denoted, as in set theory, by ∪. Union is intended to bring together all of the facts from its arguments. * For a union operation r1 ∪ r2 to be legal, we require that two conditions are held :
1. The relations r1 and r2 are of the same arity. Which means, they have the same numbers of attributes.
2. The domains of the ith attribute of r1 and the ith attribute of r2 are the same. * The resulting relation contains no duplicate tuples. * The set difference, denoted by –, is a binary operator. The result of the expression r1 – r2, is a relation obtained by including all tuples from r1 that do not appear in r2. * To apply this on two relations they must be union compatible. * The resulting relation contains no duplicate tuples. * we can rewrite the intersection as follows: r1 ∩ r2 = r1 − (r1 − r2) * After applying the intersection operator, we obtain a relation containing only those tuples from r1 which also appear as tuples in r2. We do not need to eliminate duplicate rows because the resulting relation cannot contain any.

*)If the two relations have some attributes with the same name, the names of such attributes in one of the two relations must be renamed

*)The division operation is denoted as R S and is expressed as follows (UR -US )(R) -(UR -US )(((UR -US )(R) X S) –R)
The expression on the right side of the –sign determines all tuples of R that are not associated with at least a tuple of S

* *)Find sailors who’ve reserved a red and a green boat. * Previous approach won’t work! Must identify sailors who’ve reserved red boats, sailors who’ve reserved green boats, then find the intersection (note that sid is a key for Sailors):
*) Projection eliminates duplicates by considering all the projected attributes values
*) See slide 24 in good one