UVA. Union DEPARTMENT OF COMPUTER SCIENCE. Algebra1


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1 Union Unioncompatible  two relations are unioncompatible if they have the same # of attributes and each attribute must be from the same domain  to compute R 1 R 2, R 1 and R 2 must be unioncompatible Order of relations unimportant R 1 R 2 = R 2 R 1 R 1 R 2 = {r R 1 r R 2 } <ex> R 1 (A, B) = {a 1 b 1, a 2 b 1 } R 2 (A, B) = {a 1 b 1, a 4 b 2 } R 3 = R 1 R 2 = {a 1 b 1, a 2 b 1, a 4 b 2 } Algebra1
2 Set Difference R 1 and R 2 must be unioncompatible R 1 R 2 = {r r R 1 r R 2 } Order of relation is important <ex> R 1 (A, B) = {a 1 b 1, a 2 b 1 } R 2 (A, B) = {a 1 b 1, a 4 b 2 } R 3 = R 1 R 2 = {a 2 b 1 } Algebra2
3 Intersection R 1 and R 2 must be unioncompatible R 1 R 2 = {r r R 1 r R 2 } Order of relation is unimportant <ex> R 1 (A, B) = {a 1 b 1, a 2 b 1 } R 2 (A, B) = {a 1 b 1, a 4 b 2 } R 3 = R 1 R 2 = {a 1 b 1 } Algebra3
4 Cartesian Product R 1 R 2 = {<r 1, r 2 > r 1 R 1 r 2 R 2 } <ex> R 1 (A, B) = {a 1 b 1, a 2 b 1 } R 2 (A, B) = {a 1 b 1, a 4 b 2 } R 3 = R 1 R 2 = {a 1 b 1 a 1 b 1, a 1 b 1 a 4 b 2, a 2 b 1 a 1 b 1, a 2 b 1 a 4 b 2 } <ex> Course = (C#, Cname, Dept) Class = (C#, Semester, Prof) Course Class = (C#, Cname, Dept, C#, Semester, Prof) Algebra4
5 Selection F  F: formula (or predicate)  selection criteria can be either logical or arithmetic  F (R) creates a relation consisting of tuples in R for which F(t) is true F (R) = {t t R F(t)} Selection is commutative A =a ( B =b (R)) = B =b ( A =a (R)) = A =a,b =b (R) Selection is distributive over binary operators F (R 1 R 2 ) = F (R 1 ) F (R 2 ) F (R 1 R 2 ) = F (R 1 ) F (R 2 ) F (R 1 R 2 ) = F (R 1 ) F (R 2 ) Algebra5
6 Selection Theorem: F (R 1 R 2 ) = F (R 1 ) F (R 2 ) <Proof> F (R 1 R 2 ) = {t t R 1 R 2 F(t)} = {t t R 1 t R 2 F(t)} = {t t R 1 F(t) t R 2 F(t)} = F (R 1 ) F (R 2 ) Algebra6
7 Projection A vertical subset of a relation Eliminates some attributes from a relation and then eliminate duplicate tuples, if any. <ex> Class (C#, Semester,Dept) PH1 F93 PHIL PH1 S94 PHIL CS662 S94 CS CS662 S93 CS C# (Class) = C# {PH1, CS662} Projection commutes with selection, if attributes for selection are among attributes in the set unto which X( F (R)) = F ( X(R)) if X R and F is a valid expression in X is taking place Algebra7
8 Join Central to relational database theory  used to combine related tuples from two relations into single tuples General form R <join condition > S where R=(A 1,..., A n ) and S=(B 1,..., B m ) will result in Q=(A 1,..., A n, B 1,..., B m )  Q has one tuple for each combination of tuples (one from R and one from S), whenever the combination satisfy the join condition Difference from Cartesian products  in join, only combinations of tuples satisfying the join condition appear in the result  in Cartesian product, all combinations of tuples appear Algebra8
9 Join Theta join  a join condition is of the form A i B j where A i is an attribute of R and B j of S, and A i and B j have the same domain  is one of the comparison operators: { =,, <, >, }  R S = (R S) Equijoin : selection predicate  if only comparison operator used is "=", it is equijoin  results of an equijoin always have one or more pairs of attributes that have identical values in every tuple  get rid of the second attribute: natural join Algebra9
10 Natural Join An equijoin on all attributes with the same name in two relations, followed by removal of superfluous attributes 1) compute r s 2) for each attribute A named in both R and S, select those tuples from r s where r.a = s.a (they are called join attributes) 3) for each attribute, project out s.a Join selectivity  if no combination of tuples satisfies the join condition, it results in an empty relation with 0 tuples  if r has n tuples and s has m tuples, the result of a join will be between 0 and n m tuples  join selectivity = expected size of result / n m  if no join condition to satisfy, join becomes a Cartesian product (also called a cross join) Algebra10
11 Properties of Join Join has a selecting power  to find A =a (r), define s(a) with a single tuple t such that t(a)=a and perform r s = A =a (r)  to find A =a,b =b(r), create a singleton selection relation s with schema (A, B) such that s = {ab} then r s = A =a,b =b (r) Join is commutative r s = s r Join is associative r (s q) = (r s) q Algebra11
12 Properties of Join Join and projection operators are not inverse of each other, but perform complementary functions  let q = r s and r = R (q) then R(q) = R (r s) r <ex> r: A B a1 a1 b1 b2 s: B C b1 c1 q: A B C a1 b1 c1 AB(q) = r : A B a1 b1 Algebra12
13 Properties of Join What if we reverse the order of and?  let r = R (q) and s = S (q) where q is a relation with Q = R then q r s = R (q) S(q) S Algebra13
14 Properties of Join: Example r: A B s: B C a1 b1 b1 c1 a2 b1 b1 c2 r s: A B C a1 b1 c1 a1 b1 c2 a2 b1 c1 a2 b1 c2 q: A B C a1 b1 c1 a2 b1 c2 r= R (q): A B s= S (q): B C a1 b1 b1 c1 a2 b1 b1 c2 Algebra14
15 Lossless Decomposition Relation q decomposes losslessly into schemas R and S if q = R (q) S(q) <ex> r: A B s: B C a1 b1 b1 c1 a2 b2 b2 c2 r s = q: A B C a1 b1 c1 a2 b2 c2 r = AB (q): A B a1 b1 a2 b2 Algebra15
16 Division The relation r s is a relation with schema R S such that a tuple t q = r s, if for every tuple T s s, there is a tuple T r r satisfying t r (S) = t s (S) t r (R S) = t((r S) Suitable for queries that include the phrase "for all" Division is seldom implemented in practice r s = R S (r) R S (( R S (r) s) r) Algebra16
17 Examples of Division Find all customers who have an account at all branches located in Brooklyn Branchschema (Bname, Assets, Bcity) Depositschema (Bname, A#, Cname, Balance) 1) All branches in Brooklyn r 1 = B name ( B city = Brooklyn (branch)) 2) All (Cname, Bname) pairs from deposit r 2 = B name,c name (deposit) 3) Customers in r 2 with every branch name in r 1 r 3 = r 2 r 1 = B name,c name (deposit) B name( B city = Brooklyn(branch)) Algebra17
18 Examples of Division Given the relation showing the facultystudent relationships, find the faculty who teaches every students in the given set. r: Faculty# Student# s: Student# s : Student# s": Student# r s: Faculty# r s : Faculty# r s" = Algebra18
19 Renaming A < B (r) To treat two distinct attributes as identical (for join) <ex> Class (Name, Birthdate) History (Date, Event) Who, if any, in the class was born on the day John F. Kennedy was assassinated? Class B date < Date ( Event = JFK.assassin (History)) Two attributes must have the same domain Important in practice, but not much used in theory Equijoin is similar to join with renaming Class History = (Name, Bdate, Date, Event) Bdate=Date Class B date < Date (History) = (Name, Bdate, Event) Algebra19
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