1. Computer Science 317 Database Management
The Relational Algebra (database assembly language)

2. Relational Algebra - Introduction
Just as numerical operations (+,-,*, etc.) operate on numbers and produce numbers, the relational operators act on relations (tables) and produce new relations.
With number systems, we build an algebra using the operators, constants, variables, parentheses.
So with the relational algebra.

3. The SELECT Operator Unary - operates on a single table
Result table has same attributes as operand
Predicate - A function that returns true or false.
If F is a predicate on tuples of type R and r is instance of R, then F(r) = {t  r | F(t) is true} or <sel-cond>(<relation_name>)

4. SELECT (cont.)
Legal selection conditions are formed by combining comparison clauses using AND, OR, NOT.
Comparison operators: =, <, , >, , 
Comparison clauses: <attr name> <comp op> <attr name> or <attr name> <comp op> <constant>

5. SELECT (cont.) Examples: Note: The operator “selects” rows from table.
CLASS = ‘02’(SMAJORS)
(PREREQ=‘210’) OR (PREREQ=‘211’)(COURSES)
Note: The operator “selects” rows from table.

6. The PROJECT Operator
Unary: result attributes are subset of those of the operand.
Note: Selects columns.
If we have schemas R(A1,…,An) and B={Ai1,…, Aik} with {Ai1,…,Aik}{A1,…,An } then B (r) = b where b is instance of B(Ai1,…, Aik) and tb if and only if there is ur such that t[Aij ]= u[Aij ] for j=1,…,k

7. PROJECT (cont.) <attr list> (<rel name>)
Note: must eliminate duplicates
Example r
A B C D a x s a y s c x t d x u a y t
A(A=‘a’ or D=‘u’(r))
A a d
A=‘a’ or D=‘u’(r)
A B C D a x s a y s d x u
a y t
A,D(r)
A D a s c t d u
a t

8. Set Theoretic Operations
UNION (Binary on relations of same type): r  s
INTERSECTION: r  s
DIFFERENCE: r - s
Convention: Result uses attribute names of first operand.

9. Set Theoretic (cont.) CARTESIAN PRODUCT: (Binary) r A B a 1 b 0 a 2 s
C D a c a 2
r x s
A B C D a a c a a b a c b a a a c
a a 2

10. The JOIN Operator
JOIN: r <condition> s = <condition>(r x s)
EQUIJOIN: condition uses only ‘=‘ operations
NATURAL JOIN: eliminate duplicate columns in EQUIJOIN - denoted by *
Actually, the NATURAL JOIN requires that the relations have common attributes and then selects tuples where the common attributes are the same.

11. The RENAME Operator
Sometimes it is convenient to consider a relation, s, which is the same as relation r except that the attributes have been renamed: B1,…,Bn(r) where B1,…,Bn are the new attribute names.

12. Natural Join Example Faculty Lname SSN Office Adams 111 304
Smith
Wilson
SMajors
Lname AdvSSN Class ID
Jones
Baker
Miller
Dodson
Cardwell
Faculty * SLName,SSN,Class,Id(SMajors)
Lname SSN Office SLName Class ID
Adams Jones
Adams Dodson
Adams Cardwell
Smith Baker
Wilson Miller

13. The DIVISION Operator
If we have r of type R(A1,…,An,B1,…,Bm) and s of type S(B1,…,Bm), then rs = t of type T(A1,…,An) where u  t if and only if for every tuple v  s, we have uv  r.

14. DIVISION - Example H = HasTaught FacSSN SLName ID 111 Miller 22
Smith
Miller
Wilson 44
Jones
Jones
Wilson 44
P = PhiBK
SLname ID
Jones
Miller
H  P
FacSSN
111
222
F = FullProf
FacSSN
111
333
H  F
SLname ID
Miller

15. A Complete Set of Relational Operators
Factoid: {, , , -, x} form a complete set; i.e., all other relational algebra operations can be derived from these.
Example: Derivation of division operator
r2  ((r1 x s) - r
r3  A1,…,An (r2) : all A parts from r that do not match all s parts.
rs  r1- r3
r1  A1,…,An (r) : all A parts from r

16. s = P = PhiBK
SLname ID
Jones
Miller
r1 = A1,…,An (r)
FacSSN
111
222
333
r = H = HasTaught
FacSSN SLName ID
Miller
Miller
Smith
Miller
Wilson 44
Jones
Jones
Wilson 44
r1 x s
FacSSN SLName ID
Jones
Miller
Jones
Miller
Jones
Miller
r3 = A1,…,An (r2)
FacSSN
333
H  P = r1 – r3
FacSSN
111
222
r2 = (r1 x s) - r
FacSSN SLName ID
Jones