Databases : Relational Algebra 2007, Fall Pusan National University Ki-Joune Li These slides are made from the materials that Prof. Jeffrey D. Ullman distributes via his course web page (

STEMPNU 2 What is an “ Algebra ” Mathematical system consisting of:  Operands: variables or values from which new values can be constructed.  Operators: symbols denoting procedures that construct new values from given values.

STEMPNU 3 What is Relational Algebra? An algebra  Operands: relations or variables that represent relations.  Operators: common operations that we need to do with relations in a database. Relational Algebra: a Theoretical basis of query language.

STEMPNU 4 Core Relational Algebra Set Operators  Union, intersection, and difference. Selection: picking certain rows. Projection: picking certain columns. Products and joins: compositions of relations. Renaming of relations and attributes.

STEMPNU 5 Selection R1 = SELECT C (R)  C is a condition (as in “ if ” statements) that refers to attributes of R.  R1 is all those tuples of R that satisfy C.  Denoted as  C (R)

STEMPNU 6 Example barbeerprice Joe’sBud2.50 Joe’sMiller2.75 Sue’sBud2.50 Sue’sMiller3.00 Relation Sells: JoeMenu = SELECT bar=“Joe’s” (Sells): barbeerprice Joe’sBud2.50 Joe’sMiller2.75

STEMPNU 7 Projection R1 = PROJ L (R)  L : a list of attributes from the schema of R.  R1 is constructed by looking at each tuple of R, extracting the attributes on list L, in the order specified, and creating from those components a tuple for R1.  Eliminate duplicate tuples, if any.  Denoted  L (R)

STEMPNU 8 Example Relation Sells: Prices = PROJ beer,price (Sells): barbeerprice Joe’sBud2.50 Joe’sMiller2.75 Sue’sBud2.50 Sue’sMiller3.00 beerprice Bud2.50 Miller2.75 Miller3.00

STEMPNU 9 Product (also Cartesian Product) R3 = R1 x R2  Pair each tuple t1 of R1 with each tuple t2 of R2.  Concatenation (t1 t2) is a tuple of R3.  Schema of R3: the attributes of R1 and R2, in order.  But beware attribute A of the same name in R1 and R2: use R1.A and R2.A.

STEMPNU 10 Example: R1 x R2 AB BC R1(A, B) R3(A,R1.B,R2.B,C) A,R1.B,R2.B,C R2(B, C)

STEMPNU 11 Theta-Join R3 = R1 JOIN C R2  Take the product R1 X R2.  Then apply SELECT C to the result.  Also denoted R1 C R2 As for SELECT, C can be any boolean-valued condition.  Historic versions of this operator allowed only A theta B, where theta was =, <, etc.; hence the name “ theta-join. ”

STEMPNU 12 Example Barbeerprice Joe’sBud2.50 Joe’sMiller2.75 Sue’sBud2.50 Sue’sCoors3.00 nameaddr Joe’sMaple St. Sue’sRiver Rd. BarInfo = Sells JOIN Sells.bar = Bars.name Bars BarInfo (bar, beer, price, name, addr) barbeerpricenameaddr Joe’sBud2.50Joe’sMaple St. Joe’sMiller2.75Joe’sMaple St. Sue’sBud2.50Sue’sRiver Rd. Sue’sCoors3.00Sue’sRiver Rd Sells (bar, beer,price) Bars (name,addr)

STEMPNU 13 Cartesian Product and Theta Join R1 C R2  R1 X R2 R1 C R2 =  C (R1 X R2) R1R2 X 

STEMPNU 14 Natural Join A frequent type of join connects two relations by:  Equating attributes of the same name, and  Projecting out one copy of each pair of equated attributes. Called natural join. Denoted  R3 = R1 JOIN R2 or  R3 = R1 R2 R1 JOIN R2 = Select R1.A == R2.A (R1 X R2)

STEMPNU 15 Example Barbeerprice Joe’sBud2.50 Joe’sMiller2.75 Sue’sBud2.50 Sue’sCoors3.00 Baraddr Joe’sMaple St. Sue’sRiver Rd. BarInfo = Sells Bars BarInfo (bar, beer, price, name, addr) Barbeerpriceaddr Joe’sBud2.50Maple St. Joe’sMiller2.75Maple St. Sue’sBud2.50River Rd. Sue’sCoors3.00River Rd Sells (bar, beer,price) Bars (name,addr)

STEMPNU 16 Renaming The RENAME operator gives a new schema to a relation. R1 := RENAME R1(A1, …,An) (R2) makes R1 be a relation with attributes A1, …,An and the same tuples as R2. Simplified notation: R1(A1, …,An) := R2 or  R1(A1, …,An) (R2)