1. CS 3630 Database Design and Implementation

2. Foundation of Relational Database Systems
Set Theory
Foundation of Relational Database Systems
E.F. Codd
40 minutes for S1
50 minutes for S2

3. Basic Concepts A set is a collection of elements
From a known background “Universe”
A definition we are satisfied with
Everything is in the “Universe”
An element could be any thing

4. Examples All UWP students in CS363 this semester
A = {UWP students in CS363 this semester}
B = {All UWP students who play Bridge}
Specifying the conditions of the elements in the set
C = {1, 2, 3, 4}
Listing all elements
I = {i: i is an integer}

5. Number of Elements of a Set
A = {UWP students in CS363 this semester} 56
B = {All UWP students who play Bridge} ?
C = {1, 2, 3, 4} 4
I = {i: i is an integer}

6. Cardinality
C = {1, 2, 3, 4}
|C| = 4
A = {UWP students in CS363 this semester}
|A| = 56
I = {i: i is an integer}
|I| = 
(Number of elements of finite sets)

7. No repeating elements
{1, 2, 3, 4, 2}
Not a set in classic set theory
Elements are not ordered
{1, 2, 3, 4}
{2, 4, 1, 3}
The same set

8. Empty Set A = {UWP students in CS363 this semester}
D = {s | s in A and has attended Turing
Award ceremony}
D = {}
= 
This is not empty set: {}
It’s a set with one element and the only element is an empty set

9. Sets and Elements S is a set X is an element in the “Universe”
Two possibilities:
x is in S
(x  S) or
x is not in S
(x  S)

10. Subsets For two sets A and B and any element x, if x  A then x  B
Then A is a subset of B
A  B or B  A
(similar to A < B or B > A)
A could be the same as B
A  B or B  A
(similar to A <= B or B >= A)
They are the same:  and 

11. Subsets For two sets A and B and any element x, if x  A then x  B
Then A is a subset of B
A is not a subset of B
There is an element x such that
x  A and x  B

12. Subsets
For any set X,
  X
No element e such that
e   and e  X

13. Proper Subsets For any set X,   X X  X
A is a proper subset of B, if all the three conditions are true:
A  B (A  B)
A  B
A  

14. Power Set For any set X, P(X) = {S | S is a subset of X} = {S | S  X}
C = {1, 2, 3, 4}
P(C) = {{1}, {2}, {3}, {4},
{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4},
{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4},
{1, 2, 3, 4},
{}}

15. Power Set C = {1, 2, 3, 4} P(C) = {{1}, {2}, {3}, {4},
{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4},
{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4},
{1, 2, 3, 4},
{}} }

16. The cardinality of the power set
X is a set and its cardinality is
|X|
The power set of X is P(X) and its cardinality is
|P(X)|
= 2|X|

17. Example I C = {1, 2, 3, 4} |C| = 4 P(C) = {{1}, {2}, {3}, {4},
{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4},
{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4},
{1, 2, 3, 4}, }
|P(C)| = 24 = 16

18. Example II X = {x0, x1, x2, x3, x4, x5, x6, x7} |X| = 8 |P(X)| = 28
The same as a byte with 8 bits.

19. Set Operations Set Union A  B = {x: x  A or x  B} A = {1, 2, 3, 4}
= {1, 2, 3, 4, 5}
= B  A
Range of |A  B|?
Max (|A|, |B|) <= |A  B| <= |A| + |B|

20. Set Intersection A  B = {x: x  A and x  B} A = {1, 2, 3, 4}
= {2}
= B  A
Range of |A  B| ?
0 <= |A  B| <= Min(|A|, |B|)

21. Set Difference A – B = {x: x  A but x  B} A = {1, 2, 3, 4}
= {1, 3, 4}
 B - A
B – A = {5}
Range of |A – B|?

22. A  B = {(a, b): a  A and b  B}
Cartesian Product
A  B = {(a, b): a  A and b  B}
A = {1, 2, 3, 4}
B = {2, 5}
= {(1, 2), (1, 5), (2, 2), (2, 5),
(3, 2), (3, 5), (4, 2), (4, 5)}
A  B  B  A
|A  B| = |A|  |B|

23. Cartesian Product (II)
Multiple sets A1, A2, A3, …, An
A1  A2  A3  …  An
= {(x1, x2, x3, …, xn): xi  Ai}
It is possible for some i and j, Ai = Aj.
A = {1, 2, 3, 4}
B = {2, 5}
A  B  B = {(a, b, c): a  A and b  B and c  B}
= {(1, 2, 2), (1, 5, 2), (2, 2, 2), (2, 5, 2),
(3, 2, 2), (3, 5, 2), (4, 2, 2), (4, 5, 2),
(1, 2, 5), (1, 5, 5), (2, 2, 5), (2, 5, 5),
(3, 2, 5), (3, 5, 5), (4, 2, 5), (4, 5, 5)}

24. Assignment 1 Due Wednesday (week 2), February 1 at noon
Download a copy of the assignment and re-name it as UserName_A1.doc(x)
For example, YangQ_A1.doc
Complete the assignment and drop it to
K:\Courses\CSSE\yangq\CS3630\1DropBox