1. Instructor: Mohamed Eltabakh meltabakh@cs.wpi.edu
Relational Algebra
Instructor: Mohamed Eltabakh

2. We learn next how to query the data
What we learned so far…
Entity-Relationship Model
Converting it to relational model
Relational Model
Creating table
Defining constraints
Inserting, deleting, and updating data
We learn next how to query the data

3. Query Language
Query Language
Define data retrieval operations for relational model
Express easy access to large data sets in high-level language, not complex application programs
Categories of languages
Procedural: What you want and how to get it
Non-procedural, or declarative: What you want (without how)
SQL: High-level language for relational algebra.
Relational Algebra : Operator semantics based on set or bag theory
Relational algebra form underlying basis (and optimization rules) for SQL

4. Relational Algebra Basic operators
Set Operations (Union: ∪, Intersection: ∩ ,difference: – )
Select: σ
Project: π
Cartesian product: x
rename: ρ
More advanced operators, e.g., grouping and joins
The operators take one or two relations as inputs and produce a new relation as an output
One input  unary operator, two inputs binary operator

5. Relational Algebra
Allows to build expressions using composition of the available operators
For example, arithmetic expressions are expressions of operators
(w + t) / ((x + y) * 3)
In relational algebra, instead of variables we have relations

6. Set Operators Union, Intersection, Difference
Defined only for union compatible relations
Relations are union compatible iff
they have same sets of attributes (schema), and
the same types (domains) of attributes
Example : Union compatible or not?
Student (sNumber, sName)
Course (cNumber, cName)
Not compatible

7. Union over sets: 
Consider two relations R and S that are union-compatible R S
R  S A B 1 2 3 4 5 6 A B 1 2 3 4 A B 1 2 3 4 5 6

8. Union over sets:  Notation: R ∪ S Defined as:
R ∪S = {t | t∈R or t∈S}
For R ∪S to be valid, they have to be union-compatible

9. Difference over sets: -
R – S are the tuples that appear in R and not in S
Defined as:
R – S = {t | t ∈R and t∈ S} R S
R – S A B 1 2 3 4 A B 1 2 5 6 A B 3 4

10. Intersection over sets: ∩
Consider two Relations R and S that are union-compatible R S
R ∩ S A B 1 2 3 4 A B 1 2 3 4 5 6 A B 1 2 3 4

11. Intersection: ∩ Notation: R ∩ S Defined as: Note: R ∩ S = R– (R–S)
R ∩ S = { t | t ∈ r and t ∈ s }
Note: R ∩ S = R– (R–S) R S

12. Selection: σ Select: σc (R): c is a condition on R’s attributes
Select subset of tuples from R that satisfy selection condition c
σ(C ≥ 6) (R) R A B C 1 2 5 3 4 6 7 A B C 3 4 6 1 2 7

13. Selection: σ Notation: σc(R) c is called the selection predicate
Defined as:
σc(R) = {t | t ∈ R and c(t) is true}
c is a formula in propositional calculus consisting of terms connected by :
∧ (and), ∨ (or), ¬ (not)
Each term is one of:
<attribute> op <attribute> | <attribute> op <constant>
op is one of: =,≠,>,≥.<.≤
Example of selection:
σ branch_name=“Perryridge” ^ balance>1000 (account)

14. Selection: Example R
σ ((A=B) ^ (D>5)) (R)

15. Project: π πA, C (R) R A1, A2, …, An are called Projection List
πA1, A2, …, An (R), with A1, A2, …, An  attributes AR
returns all tuples in R, but only columns A1, A2, …, An
A1, A2, …, An are called Projection List
πA, C (R) R A B C 1 2 5 3 4 6 7 8 A C 1 5 3 6 7 8

16. Cross Product (Cartesian Product): X
R X S S R

17. Cross Product (Cartesian Product): X
Notation R x S
Defined as:
R x S = {t q | t ∈ r and q ∈ s}
Assume that attributes are all unique, otherwise renaming must be used

18. Renaming: ρ ρS (R) changes relation name from R to S
ρS(A1, A2, …, An) (R) renames also attributes of R to A1, A2, …, An
ρS (R)
ρS(B->X, C, D) (R) R S S B C D 2 3 10 11 6 7 12 B C D 2 3 10 11 6 7 12 X C D 2 3 10 11 6 7 12

19. Composition of Operations
Can build expressions using multiple operations
Example: σA=C(R x S)
R X S S R
σA=C(R x S)

20. Banking Example branch (branch_name, branch_city, assets)
customer (customer_name, customer_street, customer_city)
account (account_number, branch_name, balance)
loan (loan_number, branch_name, amount)
depositor (customer_name, account_number)
borrower (customer_name, loan_number)

21. Example Queries

22. Example Queries (Cont’d)

23. Example Queries (Cont’d)

24. Example Queries (Cont’d)

25. Example Queries (Cont’d)

26. More Operators

27. What We Covered So Far…
Set Operations (Union: ∪, Intersection: ∩ ,difference: – )
Select: σ
Project: π
Cartesian product: x
rename: ρ

28. Example Queries Find customers who have neither accounts nor loans
πcustomer_name(customer) -
(πcustomer_name(borrower) U πcustomer_name(depositer))

29. Natural Join: R ⋈ S (Join on the common attributes)
Consider relations
R with attributes AR, and
S with attributes AS.
Let A = AR ∩ AS = {A1, A2, …, An} – The common attributes
In English
Natural join R ⋈ S is a Cartesian Product R X S with equality predicates on the common attributes (Set A)

30. Natural Join: R ⋈ S R ⋈ S can be defined as :
πAR – A, A, AS - A (σR.A1 = S.A1 AND R.A2 = S.A2
AND … R.An = S.An
(R X S))
Project the union of all attributes
Equality on common attributes
Cartesian Product
Common attributes appear once in the result

31. Natural Join: R ⋈ S: Example

32. Theta Join: R ⋈C S Theta Join is cross product, with condition C
It is defined as :
R ⋈C S = (σC (R X S))
Theta join can express both Cartesian Product & Natural Join R S A B 1 2 3 D C 2 3 4 5
R ⋈ R.A>=S.CS A B D C 3 2

33. Example Queries
Find customer names having account balance below 100 or above 10,000
πcustomer_name (depositor ⋈
πaccount_number(σbalance <100 OR balance > 10,000 (account)))

34. Assignment Operator: 
The assignment operation (←) provides a convenient way to express complex queries on multiple line
Write query as a sequence of line consisting of:
Series of assignments
Result expression containing the final answer
Assignment must always be made to a temporary relation variable
May use a variable multiple times in subsequent expressions
Example:
R1  (σ ((A=B) ^ (D>5)) (R – S)) ∩ W
R2  R1 ⋈(R.A = T.C) T
Result  R1 U R2

35. Example Queries
Find customers having account balance below 100 and loans above 10,000
R1  πcustomer_name (depositor ⋈ πaccount_number(σbalance <100 (account)))
R2  πcustomer_name (borrower ⋈ πloan_number(σamount >10,000 (loan)))
Result  R1 ∩ R2

36. More Relational Operators

37. Outer Join
An extension of the join operation that avoids loss of information
Computes the join and then adds tuples form one relation that does not match tuples in the other relation to the result
Uses null values to fill in empty attributes with no matching
Types of outer join between R and S
Left outer (R o⋈ S): preserve all tuples from the left relation R
Right outer (R ⋈o S): preserve all tuples from the right relation S
Full outer (R ⋈ S): preserve all tuples from both relations o

38. Left Outer Join (R o⋈ S): Example
R ⋈ S
(R o⋈ S)

39. Right Outer Join (R ⋈o S): Example
R ⋈ S
(R ⋈o S)

40. Full Outer Join (R ⋈ S): Example

41. Duplicate Elimination:  (R)
Delete all duplicate records
Convert a bag to a set R
 (R) A B 1 2 3 4 A B 1 2 3 4

42. Extended Projection: πL (R)
Standard project
L contains only column names of R
Extended projection
L may contain expressions and assignment operators
π C, VA, X C*3+D (R)

43. Grouping & Aggregation operator: 
Aggregation function takes a collection of values and returns a single value as a result
avg: average value
min: minimum value
max: maximum value
sum: sum of values
count: number of values
Grouing & Aggregate operation in relational algebra
g1,g2, …gm, F1(A1), F2(A2), …Fn(An) (R)
R is a relation or any relational-algebra expression
g1, g2, …gm is a list of attributes on which to group (can be empty)
Each Fi is an aggregate function applied on attribute Ai within each group

44. Grouping & Aggregation Operator: ExampleS R
branch_name,sum(balance)(S)
sum(c)(R)

45. Summary of Relational-Algebra Operators
Set operators
Union, Intersection, Difference
Selection & Projection & Extended Projection
Joins
Natural, Theta, Outer join
Rename & Assignment
Duplicate elimination
Grouping & Aggregation

46. Example Queries Find customer names having loans with sum > 20,000
πcustomer_name (σsum > 20,000 (customer_name, sum  sum(amount)(loan ⋈ borrower)))

47. Example Queries
Find the branch name with the largest number of accounts
R1  branch_name, countAccounts  count(account_number)(account)
R2  Max  max(countAccounts)(R1)
Result  πbranch_name(R1 ⋈countAccounts = Max R2)

48. Example Queries
Find account numbers and balances for customers having loans > 10,000
πaccount_number, balance
( (depositor ⋈ account) ⋈
(πcustomer_name (borrower ⋈ (σamount >10,000 (loan)))) )

49. Reversed Queries (what does it do)?
πcustomer_name(customer) - πcustomer_name(borrower)
Find customers who did not take loans

50. Reversed Queries (what does it do)?
R1  (MaxLoan  max(amount)(σbranch_name= “ABC” (loan)))
Result  πcustomer_name(borrower ⋈ (R1 ⋈MaxLoan=amount^branch_name= “ABC” loan))
Find customer name with the largest loan from a branch “ABC”

51. Exercise 1
For branches that gave loans > 100,000 or hold accounts with balances >50,000, report the branch name

52. Exercise 2
Find customer name with the largest loan from a branch in “NY” city