# UVA. Union DEPARTMENT OF COMPUTER SCIENCE. Algebra-1

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1 Union Union-compatible - two relations are union-compatible 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 union-compatible 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 } Algebra-1

2 Set Difference R 1 and R 2 must be union-compatible 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 } Algebra-2

3 Intersection R 1 and R 2 must be union-compatible 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 } Algebra-3

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) Algebra-4

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 ) Algebra-5

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 ) Algebra-6

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 Algebra-7

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 Algebra-8

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 Algebra-9

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) Algebra-10

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 Algebra-11

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 Algebra-12

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 Algebra-13

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 Algebra-14

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 Algebra-15

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) Algebra-16

17 Examples of Division Find all customers who have an account at all branches located in Brooklyn Branch-schema (B-name, Assets, B-city) Deposit-schema (B-name, A#, C-name, Balance) 1) All branches in Brooklyn r 1 = B name ( B city = Brooklyn (branch)) 2) All (C-name, B-name) 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)) Algebra-17

18 Examples of Division Given the relation showing the faculty-student 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" = Algebra-18

19 Renaming A < B (r) To treat two distinct attributes as identical (for join) <ex> Class (Name, Birth-date) 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, B-date, Date, Event) B-date=Date Class B date < Date (History) = (Name, B-date, Event) Algebra-19

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