Relational Algebra Part 1. Definitions.

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1 .. Cal Poly pring 2016 CPE/CC 365 Introduction to Database ystems Alexander Dekhtyar Eriq Augustine.. elational Algebra Notation, T,,... relations. t, t 1, t 2,... tuples of relations. t (n tuple with n attributes. elational Algebra Part 1. Definitions. t[1], t[2],..., t[n] 1st, 2nd,... nth attribute of tuple t. t.name attribute name of tuple t (attribute names and numbers are interchangeable..name, [1],... attribute name or attribute number 1 (etc. of relation. (X 1 : V 1,..., X n : V n relational schema specifying relation with n attributes X 1,..., X n of types V 1,..., V n. We let V i specify both the type and the set of possible values in it. Base Operations Union Definition 1 Let = {t} and = {t } be two relations over the same relational schema. Then, the union of and, denoted is a relation T, such that: T = {t t or t }. Union of two relations combines in one relation all their tuples. Note: it is important to note that for union to be applicable to two relations and, they must have the same schema. Difference Definition 2 Let and be two relations over the same relational schema. Then, the difference of and, denoted is a relation T such that: T = {t t and t }. Difference of two relations and is a relation that contains all tuples from that are not contained in. 1

2 Cartesian Product Definition 3 Let and be two relations. The Cartesian product of and, denoted is a relation T such that: T = {t t = (a 1,..., a n, b 1,..., b m (a 1,..., a n (b 1,..., b m }. Cartesian product glues together all tuples of one relation with all tuples of the other one. election Definition 4 Let database D consist of relations 1,... n. Let X, Y be attributes of one of the relations 1,... n. An atomic selection condition is an expression of one of the forms: X op α; X op Y, where α V and V is X s domain; op {=,, <, >,, } and X and Y have comparable types. An atomic selection condition is a selection condition. Let C 1 and C 2 be two selection conditions. Then C 1 C 2, C 1 C 2 and C 1 are selection conditions. An expression is a valid selection condition only if it can be constructed using the procedure above. election conditions specify information needs of the user. Definition 5 Let t = (a 1,..., a n be a relational tuple. Let C = X op α and C = Xop Y be two atomic selection conditions. t satisfies C iff t.x op α is a true statement. t satisfies C iff t.x op t.y is a true statement. t satisfies C 1 C 2 iff t satisfies both C and C. t satisfies C 1 C 2 iff t satisfies either C 1 or C 2. t satisfies C 1 iff t does not satisfy C 1. The notion of satisfaction allows us to distinguish between the tuples that contain information sought by the user and those that do not. Definition 6 Let be a relation and C be a selection condition. election on with C denoted σ C ( is a relation T such that: T = {t t t satisfies C} Projection Definition 7 Let be a relation with schema (X 1 : V 1,..., X n : V n. Let F be a sequence Y 1,..., Y m of attributes from. Then projection of on F denoted π F ( is a relation T such that T = {t t = (a 1,..., a m, ( t ( 1 i ma i = t.y i } Projection throws out some attributes of the relation and possibly rearranges the order of the remaining ones. 2

3 ename This is a utility operation, which gives a new name to a given relation and (optionally renames its attributes. Three possible ways in which it is used are: ρ ( : relation is now renamed to. Attributes preserve their names. ρ (A1,...,A n(: relation is now renamed to. Its attributes are now named A 1,..., A n. ρ (A1,...,A n(: relation keeps its name, but the attributes are renamed to A 1,..., A n. Note: enaming a relation is useful in situations when you need to write a relational algebra expression that involves using one relation several times for different purposes. It can be combined with the projection operation s power to rearrange the order of columns. More on Conditions A more generalized version of conditions used in selection and join operations may be described as follows. Let database D consist of relations 1,... k. Let X 1,... X N be all attribute names used in D. A valid identifier is one of the following: X,.Y, c where X, Y are attribute names, Y is an attribute of relation, and c is a constant. An atomic condition is an expression of the form f(x 1,..., X m opg(y 1,..., Y k, where op {=,, >, <,, }, X 1,..., X m, Y 1,... Y k are all valid identifiers and f (m (. and g (k (. are computable functions. Finally, atomic conditions are conditions and if C 1 and C 2 are conditions than so are C 1 C 2, C 1 C 2 and C 1. Valid conditions are only those that can be constructed using the procedures described above. This definition of produces a larger set of conditions, some of which are no selection conditions. It also extends the set of selection conditions: e.g.,.x 1 +.X 2 (.X is a valid selection constraint now. Derived Operations ome important relational algebra operations can be derived from the basic operations. Intersection Definition 8 Let and be two relations with the same relational schema. An intersection of and, denoted is a relation T such that T = {t t and t } Intersection finds tuples common to two relations. From set theory we know that = (, hence, intersection is a derived operation in our relational algebra. 3

4 Division (Quotient Definition 9 Let be a relation of arity r with relational schema X 1 : V 1,..., X r : V r and be a relation of arity s < r (s 0 with schema Y 1 : V r s+1,..., Y s : V r. The result of division of on (also known as the quotient of and, denoted /, is a relation T of arity r s with schema X 1 : V 1,..., X r s : V r s such that T = {t = (a 1,..., a r s ( s = (a r s+1,..., a r ((a 1,..., a r } Division finds all prefixes in one relation that have every suffix from the second relation. Division can be derived as follows: Let F = X 1,..., X r s. Then, Joins / = π F ( π F ((π F (. The family of join operations allows us to intelligently combine together the contents of two (or more different relations. Θ-Join Also known as condition join. Definition 10 Let and be two relations with attributes (X 1,..., X n and (Y 1,..., Y m respectively and let C =.XΘ.Y be a join condition, with Θ {=,, <, >,, }. A Θ-join of and, denoted C, is a relation T such that: T = {t = (a 1,..., a n, b 1,..., b m r = (a 1,..., a n s = (b 1,..., b m s.xθr.y is true }. Join selects the tuples from the Cartesian product of two relations that match a given constraint (typically, comparing the attributes from both relations. Θ-join can be expressed via selection and Cartesian product as follows: Equijoin C = σ C (. Equijoin is a special type of Θ-join where the join condition is a conjunction of equalities:.x =.Y. Natural Join Natural Join is a type of equijoin of two relations. Definition 11 Let (X 1,..., X n and (Y 1,..., Y m be two relations and let Z 1,... Z k all common attributes of and. Let Y = (Y 1,..., Y m (Z 1,..., Z k. A natural join of and, denoted is a relation T such that T = π ( X1,...,X n,y.z 1=.Z 1...Z k =.Z k Natural join looks at all common attributes of two relations and joins on them, removing one set of common attributes from the final relation. 4

5 emijoin A semijoin of two relations performs a(ny join operation of two relations and then projects the result onto the fields of one of the relations: Query Trees Θ = π ( θ Θ = π ( θ Just like in traditional math, relational algebra expressions can get pretty complex. To help break expressions down, we can use Query Trees. Just like syntax trees (the normal algebra equivalent, we can use a tree like structure to break down an expression into its sub-expressions. Below is an example of a simple expression and tree (for both algebra and relational algebra: yntax Tree Query Tree x + y z T + x T y z Each parent is an operator and its children are the operation s operands. This means that binary operators (e.g. + and will have two children, while unary operators (e.g. (negation and π will only have one child. Below are examples of simple queries using the different relational algebra operators. σ b=5 ( π a,b ( ρ ( C Θ σ b=5 π a,b ρ C Θ Here is a more complex example using the BAKEY dataset: Find all the customers (last name, first name who purchased a cherry tart or a cake on October 30, 2007, but who has never purchased any cookies. (The strange formatting is not normally part of relational algebra, this expression is just too big to fit on a single line. π Customers.lastName,Customers.firstName ( Customers C.id=Customers.cId ρ C(id ( π eceipts.customer ( σ food= cake (food= tart flavor= cherry (Goods Goods.gId=Items.item ( Items Item.receipt=eceipts.rNumber σ saledate= (eceipts π eceipts.customer ( eceipts eceipts.rnumber=items.receipt (Items Items.item=Goods.gId σ food= cookie (Goods 5

6 π Customers.lastName,Customers.firstName C.id=Customers.cId ρ C(id Customers π eceipts.customer Goods.gId=Items.item π eceipts.customer eceipts.rnumber=items.receipt eceipts Items.item=Goods.gId σ food= cake (food= tart flavor= cherry Goods Item.receipt=eceipts.rNumber σ saledate= Items eceipts Items σ food= cookie Goods Examples Consider the following five relations (W is at the bottom of the page: : A B C D : B E F T: 1 1 a a 1 a c 1 2 a b 1 b a 2 1 b b 2 a a 2 2 b b 3 b b 3 b a a 2 c a Here are the results of some relational algebra operations on these tables. V: 3 a 2 3 b 1 1 c 0 T V 3 b 2 3 a 2 3 b 1 1 c 0 σ A=2 ( A B C D 2 1 b b 2 2 b b 2 2 a a T V 3 b 2 σ E b ( B E F 1 a c 2 a a 2 c a V T 3 a 2 3 b 1 1 c 0 σ G>2 (T V T σ G>A (T 6

7 π A,D ( A D 1 a 1 b 2 b 2 a π F,B ( F B c 1 a 1 a 2 b 3 W: /W A B C D 2 1 b b 2 2 T W T. W.A B b b T T.A=V.G V T.A T.E T.G V.A V.E V.G 3 a 2 3 b 1 3 a 2 3 b 1 T V T B E F 1 a c 1 b a 2 a a 3 b b π E (T E b a W T W.A B T b b π E,G,A (T E G A b 1 2 a 2 3 b 1 3 a 2 1 c 0 1 T A B C D E G 1 1 a a a a a b a b a a b b b b b b b b b b b b b b a a b a a b 4 T 3 b 2 B>A T B.E F A T.E G 2 a a 2 a a 3 b b 3 b b 3 b b 3 b b 1 B 6 2 c a 2 c a V B E F A G 1 a c a c b a b a a a a a b b b b c a 1 0 7