1. STRUCTURE OF PRESENTATION :
1. Basic database concepts **
Entr’acte
2. Relations 8.
SQL tables
3. Keys and foreign keys 9.
SQL operators I
4. Relational operators I
10. SQL operators II
Relational operators II 11.
Constraints and predicates 12.
The relational model A.
SQL constraints
SQL vs.
the relational model References
Copyright C. J. Date 2013

2. OK ... Now we can start using the DB for real work!
To be specific, we can issue “queries” (retrieval requests), using
operators of the relational algebra
/* see next page */
Copyright C. J. Date 2013

3. CODD’S ORIGINAL RELATIONAL ALGEBRA : AN OVERVIEW
product
restrict project
a b c
x y a x y b c
intersect
union
difference
(natural) join a1 b1 c1 a2 b2 c2 a3 b3 c3 a1
a2 a3 b1
b1 b2 c1
c1 c2
Copyright C. J. Date 2013

4. ASIDE :
Actually “Codd’s original algebra” included another operator, viz. relational division
Deliberately omitted here, because:
There’s a better approach to “division style” queries,
using image relations (see later)
Relational division didn’t solve the problem anyway
Copyright C. J. Date 2013

5. Now we can start using the DB for real work!
To be specific, we can issue “queries” (retrieval requests), using operators of the relational algebra
Important: Operators are all READ-ONLY and SET LEVEL
Important: Relations form a CLOSED SYSTEM under the operators of the algebra !!!
I.e., result of every relational operation is ANOTHER RELATION … so we can write nested relational expressions
/* analogy with arithmetic */
Copyright C. J. Date 2013

6. RESTRICT :
Get part information for parts whose weight is less than 12.5:
P WHERE WEIGHT < 12.5
Result has same heading as P and body = tuples of P for which specified restriction condition evaluates to TRUE:
PNO
PNAME
COLOR
WEIGHT
CITY
P1 P5
Nut Cam
Red Blue
12.0
London Paris
Copyright C. J. Date 2013

7. PROJECT : For each part, get part number, color, and city:
P { PNO , COLOR , CITY } or P { ALL BUT PNAME , WEIGHT }
Result has heading as specified and body = corresponding
(sub)tuples of P:
PNO
COLOR
CITY P1
Red
London P2
Green
Paris P3
Blue
Oslo P4 P5 P6
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8. ASIDE : Let A be a set; then every subset of A is a set.
It follows that:
Every subset of a tuple is a tuple (“every subtuple is a tuple”)
Every subset of a heading is a heading
Every subset of a body is a body /* see restriction */
Copyright C. J. Date 2013

9. PROJECT (cont.) : What part color / city combinations currently exist?
P { COLOR , CITY } or P { ALL BUT PNO , PNAME , WEIGHT }
Result has heading as specified and body = corresponding (sub)tuples of P:
“Duplicates eliminated” !!!
(Duplicate elimination occurred in the previous example too, of course, but had no effect)
COLOR
CITY
Red
London
Green
Paris
Blue
Oslo
Copyright C. J. Date 2013

10. CLOSURE : AN EXAMPLE
Get part number, color, and city for parts whose weight is less than 12.5:
( P WHERE WEIGHT < 12.5 ) { PNO , COLOR , CITY }
Note use of parentheses (so restriction done first, then projection)
/* attribute
order
insignificant */
PNO
CITY
COLOR
P1 P5
London Paris
Red Blue
Copyright C. J. Date 2013

11. RESTRICT AND PROJECT : Tutorial D syntax (restrict and project):
rx WHERE bx
boolean expression that can be evaluated for any tuple t of r by examining t in isolation
— actually Tutorial D allows bx to be
arbitrarily complex /* so does SQL */
relational expression
denoting relation r
rx { [ ALL BUT ] attribute name commalist }
relational expression subset of heading of r
denoting relation r
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12. S { CITY } UNION P { CITY } CITY
Operands must be of same type (i.e., have same heading); result is of same type also
Get cities such that at least one supplier or at least one part is
located in that city:
S { CITY } UNION P { CITY } CITY
/* note appeal to closure */
Athens London Oslo Paris
“Duplicates eliminated” !!!
Copyright C. J. Date 2013

13. UNION (cont.) :
Operands must be of same type (i.e., have same heading);
result is of same type also
Suppose parts have extra attribute STATUS, of type INTEGER
Get status / city pairs such that at least one supplier or part has that combination:
P { STATUS , CITY } UNION S { CITY , STATUS }
/* attribute order insignificant */
/* operand order ditto */
Copyright C. J. Date 2013

14. INTERSECT :
Operands must be of same type (i.e., have same heading); result is of same type also
Get cities such that at least one supplier and at least one part are both located in that city:
S { CITY } INTERSECT P { CITY } CITY
London
Paris
“Duplicates eliminated” ???
Copyright C. J. Date 2013

15. INTERSECT (cont.) :
Operands must be of same type (i.e., have same heading); result is of same type also
Suppose parts have extra attribute STATUS, of type INTEGER
Get status / city pairs such that at least one supplier and at
least one part both have that combination:
S { STATUS , CITY } INTERSECT P { CITY , STATUS }
/* attribute order insignificant */
/* operand order ditto */
Copyright C. J. Date 2013

16. DIFFERENCE :
Operands must be of same type (i.e., have same heading); result is of same type also
Get cities such that at least one supplier is located in that city and no part is:
S { CITY } MINUS P { CITY } CITY
Athens
“Duplicates eliminated” ???
Copyright C. J. Date 2013

17. DIFFERENCE (cont.) :
Operands must be of same type (i.e., have same heading);
result is of same type also
Suppose parts have extra attribute STATUS, of type INTEGER
Get status / city pairs such that at least one supplier has that combination and no part does:
S { STATUS , CITY } MINUS P { CITY , STATUS }
/* attribute order insignificant */
Copyright C. J. Date 2013

18. UNION, INTERSECT, MINUS :
Tutorial D syntax (UNION, INTERSECT, DIFFERENCE):
rx op ry UNION or INTERSECT or MINUS
relational expressions
denoting relations of same type
Alternatively (“n-adic” version):
op { relational expression commalist } UNION or INTERSECT (not MINUS!)
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19. UNION bis :
Operands must be of same type (i.e., same attribute-name / type-name pairs) …So what about:
Get names such that at least one supplier or at least one part
has that name:
/* assume this query makes sense !!! */
Must rename at least one of the two “name” attributes before we can do
the UNION … E.g.,
( ( S RENAME { SNAME AS NAME } ) { NAME } )
UNION
( ( P RENAME { PNAME AS NAME } ) { NAME } )
/* I’ve renamed both attributes for symmetry … overall result heading has single attribute, called NAME, of type CHAR */
Copyright C. J. Date 2013

20. RENAME : S RENAME { CITY AS SCITY }
Result identical to current value of S except for renaming
SNO
SNAME
STATUS
SCITY S1
Smith 20
London S2
Jones 10
Paris S3
Blake 30 S4
Clark S5
Adams
Athens
Needed primarily as
a preliminary to either
UNION or JOIN
Tutorial D syntax (RENAME):
rx RENAME { renaming commalist }
Copyright C. J. Date 2013

21. EXERCISES :
1. We have a formal reason for prohibiting duplicate tuples in relations (what is it?), but can you think of a pragmatic reason?
Copyright C. J. Date 2013

22. EXERCISES (cont.) :
What do the following expressions denote? You can assume that projection has higher precedence than the other relational operators.
P { PNO } MINUS ( SP WHERE SNO = 'S2' ) { PNO }
( S { CITY } INTERSECT P { CITY } ) UNION P { CITY }
S { SNO } MINUS ( S { SNO } MINUS SP { SNO } )
SP MINUS SP
S INTERSECT S
( S WHERE CITY = 'Paris' ) UNION ( S WHERE STATUS = 20 )
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23. EXERCISES (cont.) :
Write Tutorial D expressions for the following queries:
Get all shipments
Get part numbers for parts supplied by supplier S2
Get suppliers with status in the range 15 to 25 inclusive
Get supplier numbers for suppliers with a status lower than that of supplier S2
Get part numbers for parts supplied by all suppliers in Paris
Copyright C. J. Date 2013

24. EXERCISES (cont.) :
You really ought to try some queries of your own ...
Copyright C. J. Date 2013

25. CODD’S ORIGINAL RELATIONAL ALGEBRA : AN OVERVIEW
product
restrict project
a b c
x y a x y b c
intersect
union
difference
(natural) join a1 b1 c1 a2 b2 c2 a3 b3 c3 a1
a2 a3 b1
b1 b2 c1
c1 c2
Copyright C. J. Date 2013

26. (Natural) JOIN : EXAMPLE
For each shipment, get part number and quantity and
corresponding supplier information:
SP JOIN S
SNO
PNO
QTY
SNAME
STATUS
CITY S1 P1
300
Smith 20
London P2
200 ..
...
..... S4 P5
400
Clark 20
London
Copyright C. J. Date 2013

27. For each shipment, get part number and quantity and
corresponding supplier information:
SP JOIN S
(Important!): Operands must be JOINABLE ... Definition:
Relations are joinable if and only if attributes with the same name are of the same type
I.e., set theory union of headings is a legal heading
page ... concept relevant to other ops as well as join */
/* see next
Copyright C. J. Date 2013

28. “Duplicates eliminated”! ... by definition
Let A and B be sets. Then their (set theory) union is the set of all elements x such that x appears in A or B or both.
“Duplicates eliminated”! ... by definition
Copyright C. J. Date 2013

29. JOIN (cont.) : For each shipment, get part number and quantity and
corresponding supplier information:
SP JOIN S
Result heading = set theory union of headings of SP and S ... Result body = set of all tuples t where t is the set theory union of a tuple from SP and a tuple from S
SNO
PNO
QTY
SNAME
STATUS
CITY S1 P1
300
Smith 20
London P2
200 ..
...
..... S4 P5
400
Clark 20
London
Copyright C. J. Date 2013

30. “Duplicates eliminated”! ... by definition In the definition of join:
Let A and B be sets. Then their (set theory) union is the set of all elements x such that x appears in A or B or both.
“Duplicates eliminated”! ... by definition
In the definition of join:
“Set theory union of headings”: OK, headings are definitely sets
“Set theory union of tuples”: Yes, tuples are sets too!
Copyright C. J. Date 2013

31. ANOTHER EXAMPLE :
Get part and supplier information for parts and suppliers that
are colocated (i.e., have the same city):
P JOIN S
Result heading = set theory union of headings of P and S ... Result body = set of all tuples t where t is the set theory union of a tuple from P and a tuple from S
PNO
PNAME
COLOR
WEIGHT
CITY
SNO
SNAME
STATUS P1
Nut
Red
12.0
London S1
Smith 20 ..
...
....
......
..... P6
Cog
Red
19.0
London S4
Clark 20
Copyright C. J. Date 2013

32. JOIN : Tutorial D syntax (JOIN): rx JOIN ry
Note: The join is always “on the basis of common attributes”
... thus, attribute renaming might sometimes be required
Alternatively (“n-adic JOIN”):
JOIN { relational expression commalist }
Copyright C. J. Date 2013

33. (Cartesian) PRODUCT :
Get all current supplier number / part number pairs:
S { SNO } TIMES P { PNO }
Result heading = set theory union of headings of S{SNO} and P{PNO} ... Result body = set of all tuples t where t is the set theory union of a tuple from S{SNO} and a tuple from P{PNO} …
Just as for JOIN, in fact, except that for TIMES the input relations must have no attribute names in common
TIMES is a special case of JOIN !!!
Copyright C. J. Date 2013

34. ... and set theory union of headings is indeed a legal heading
Recall: Relations are joinable if and only if attributes with the same name are of the same type … So relations with no common attribute names are certainly joinable !!! ...
/* certainly there aren’t any attributes with the same name that aren’t
of the same type */
... and set theory union of headings is indeed a legal heading
rx TIMES ry
Alternatively (“n-adic TIMES”):
TIMES { relational expression commalist }
Copyright C. J. Date 2013

35. ANOTHER EXAMPLE :
Get SNO / PNO pairs such that supplier SNO doesn’t supply part PNO:
( S { SNO } TIMES P { PNO } )
MINUS
( SP { SNO , PNO } )
Or:
( S { SNO } JOIN P { PNO } )
/* TIMES is supported primarily for psychological reasons */
Copyright C. J. Date 2013

36. WHAT’S MORE : INTERSECT is a special case of JOIN as well !!! Recall:
Get cities such that at least one supplier and at least one part are both located in that city:
S { CITY } INTERSECT P { CITY }
CITY
could be JOIN
London
Paris
/* INTERSECT is supported primarily for psychological reasons */
Copyright C. J. Date 2013

37. WHICH OPERATORS ARE PRIMITIVE ???
So INTERSECT and TIMES can be defined in terms of join
... i.e., not all operators are primitive
Difference between primitive and useful !!!
One possible primitive set: restrict
project join union
difference
/* could replace by semidifference … see later */
But what about rename?
/* again, see later */
Aside: In fact it’s possible to define a version of the algebra that has
just two primitives!
Copyright C. J. Date 2013

38. RELATIONAL COMPARISONS* :
Must be able to test relations for equality, of course E.g.:
S { CITY } = P { CITY } /* FALSE */
Other useful comparison ops:
    
relational inclusion
Useful shorthands:
IS_EMPTY ( rx )
IS_NOT_EMPTY ( rx )
* Are these algebraic operators as such ???
Copyright C. J. Date 2013

39. NOW I CAN EXPLAIN … R := rx /* relational assignment—generic form */
“INSERT R rx” is shorthand for:
R := R UNION rx
But what about “inserting a tuple that already exists” ??? “D_INSERT R rx” is shorthand for:
R := R D_UNION rx
“disjoint union”
Copyright C. J. Date 2013

40. “DELETE R rx” is shorthand for:
R := R MINUS rx
But what about “deleting a tuple that doesn’t exist” ??? ...
“I_DELETE R rx” is shorthand for:
R := R I_MINUS rx
“included minus”
“DELETE [ R ] R WHERE bx” is shorthand for:
R := R WHERE NOT ( bx )
Copyright C. J. Date 2013
page 100

41. “UPDATE R WHERE bx : { ... }” is shorthand too …
attribute assignment commalist
… but the details are a little complicated, so I’ll skip them
Copyright C. J. Date 2013

42. EXERCISES :
What do the following expressions denote? You can assume that projection has higher precedence than the other relational operators.
( S { SNO , CITY } JOIN P { PNO , CITY } ) { PNO , SNO }
JOIN { S , SP , P }
JOIN { ( S RENAME { CITY AS SC } ) { SC } ,
( P RENAME { CITY AS PC } ) { PC } }
SP { SNO } I_MINUS S { SNO }
( S WHERE STATUS = 20 ) { CITY } D_UNION
( P WHERE COLOR = 'Blue' ) { CITY }
Copyright C. J. Date 2013

43. EXERCISES (cont.) : Write Tutorial D expressions for the following:
Get part numbers for parts supplied by a supplier in
London
Get part numbers for parts not supplied by any supplier in London
Get pairs of part numbers such that some supplier supplies both of the indicated parts
Copyright C. J. Date 2013

44. EXERCISES (cont.) :
Again, you really ought to try some queries of your own ...
Copyright C. J. Date 2013