1. CS1022 Computer Programming & Principles
Lecture 2
Relations

2. Plan of lecture Equivalence relations Set partition Partial order
Total order
Database management systems
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3. Equivalence relation A relation R on A which is
Reflexive (x  A  (x, x)  R),
Symmetric ((x, y)  R)  ((y, x)  R)), and
Transitive (((x, y)  R and (y, z)  R)  (x, z)  R)
is called an equivalence relation
Equivalence relation generalises/abstracts equality
Pairs share some common features
Example:
Relation “has same age” on a set of people – related people belong to “same age sub-group”
Natural way to split up elements (partition)
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4. Set partition (1)
Underlying set A of equivalence relation R can be split into disjoint (no intersection) sub-sets
Elements of subsets are all related and equivalent to each other
“Equivalent” is well-defined (and can be checked)
Very important concept
Underpins classification systems
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5. Set partition (2)
A partition of a set A is a collection of non-empty subsets A1, A2, , An, such that
A  A1  A2    An
i, j, 1  i  n, 1  j  n, i  j, Ai  Aj  
Subsets Ai, 1  i  n, are blocks of the partition
Sample Venn diagram: 5 blocks A
Blocks do not overlap
Their intersection is empty A1 A2 A3 A4 A5
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6. Set partition (3) Subsets/partitions consist of related elements of A
Equivalence class Ey, of any y  A, is the set
Ey  z  A : z R y
Theorem:
Let R be an equivalence relation on a non-empty set A
The distinct equivalence classes form a partition of A
Proof in 4 parts:
Show that equivalent classes are non-empty subsets of A
Show that if x R y then Ex  Ey
Show that A  Ex1  Ex2    Exn
Show that for any two Ex and Ey, Ex  Ey  
Ey is the set of things that are in the equivalence class with respect to y. See later.
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7. Set partition (4) Example: relation R on R (real numbers) defined as
x R y if and only if x  y is an integer (Z).
x  y is an integer is an equivalence relation.
(x  y)  Z
Proof:
Since x  x  0 for any real number x, then R is reflexive
If x  y is an integer then y  x  (x  y) is the negative of an integer, which is an integer, hence R is symmetric
If x  y and y  z are integers then x  z  (x  y) + (y  z) is the sum of two integers, which is another integer, so R is transitive
Therefore R is an equivalence relation
Uses a bit of number theory. Examples: 7-5 = -(5-7); (7-5)+(5-2) = (7-2).
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8. Set partition (5) Our previous example is
y  R, Ey  z  R : (z  y)  Z
We can obtain the following equivalence classes:
E0  z  R : (z  0)  Z  Z (all integers)
E½  z  R : (z  ½)  Z  , 1½, ½, ½, 1½, 2½, 
E2 z  R : (z  2)  Z  , 1  2, 2, 1  2, 2  2, 
How would it work if we have a set of ordered pairs of names and ages, e.g. {(Jill, 35), (Bill, 37), (Phil, 35) (Mary, 35) (Will, 37)}, and the relation is 'same age as'?
R is the set of reals. Z is the set of integers.
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9. Partial order (1) "is taller than" a partial order?
A relation R on A which is
Reflexive (x  A  (x, x)  R),
Anti-symmetric ((x, y)  R) and (x  y))  ((y, x)  R)), and
Transitive (((x, y)  R and (y, z)  R)  (x, z)  R)
Is a partial order
Sets on which partial order is defined are “posets”
Partial orders help us defining precedence
Decide when an element precedes another
Establish an order among elements
Sample partial orders
“” on the set R of real numbers
“” on subsets of some universe set
"is taller than" a partial order?
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10. Partial order (2)
If R is a partial order on A and x R y, x  y, we call
x a predecessor of y
y a successor of x
An element may have many predecessors
If x is a predecessor of y and there is no z, x R z and z R y, then x is an immediate predecessor of y
We represent this as x  y
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11. Partial order (3)
Immediate predecessors graphically represented as a Hasse diagram:
Vertices are elements of poset A
If x  y, vertex x is placed below y and joined by edge
A Hasse diagram has complete information about original partial order
Provided we infer which elements are predecessors of others, by working our way upwards on chain of edges
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12. Partial order (4) Example: relation “is a divisor of” on set
Defines a partial order
Table of predecessors & immediate predecessors
element
predecessors
immediate 1
none 2 3 6
1, 2, 3
2, 3 12
1, 2, 3, 6 18 6 12 18 3 2 1
Why the branch?
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13. Total order
A total order on a set A is a partial ordering in which every pair of elements are related.
The Hasse diagram of a total order is a long chain
Examples of total orders:
Relation “” on Real numbers
Lexicographical ordering of words in a dictionary
In computing:
Sorting algorithms require total ordering of elements
Posets with minimal/maximal elements useful too
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14. Database management systems (1)
Data with lifespan
Data stored in variables only exist while program runs
Data shown on screen only exist while being shown
Some data must be persistent
It should be kept for hours, days, months or even years
Simple way to make data persistent: files
Data saved electronically on your hard-disk
Records (lines) contain numbers, strings, etc.
A “flat file” (e.g. a lexicon) allows sequential processing of records:
To access line/record n, we must read previous n1 lines
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15. Database management systems (2)
Alternative to “flat files”: databases
Databases allow efficient access to records
We can directly access record n
We can search for records which meet some criteria
Database management system (DBMS)
Means to create and manage databases (notice plural)
Infra-structure to organise/store data, records
Means to access/query records
No need to create our own database systems
Existing solutions – stable technology, free, efficient, scale up
MySQL (
Trend: databases in the “cloud” (Amazon)
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16. Database management systems (3)
Data in DB must be thought out
Only data needed should be there
Same data should not appear in two (or more) places
If data can be obtained from other data, then it should not be stored (e.g., unit price and total batch price)
Challenge of database design/management
Selecting the “right” data
Organising data (what should go with what and where)
Deciding when to re-design databases (due to problems)
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17. Tables (1) Data in DB stored as tables
Table T1: Personal details (student record system)
ID No.
Name
Sex
DOB
Marital
Status
Address
Jones
Female
1.2.83
Single
2 The Motte, Newton
Singh
Male
4.5.84
Married
4A New Road, Seaforth
Smith
17 The Crescent, Seaforth
21 Pudding Lane, Witham
Ching
Grant
9.7.83
18 Iffley Road, Reading
McKay
133 Uff Road, Reading
French
11 Finn Road, Newtown
ID. Numbers should be unique
Names can be repeated (e.g., Smith)
Problems when listing students just by name
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18. Tables (2)
A table with n columns A1, A2, , An is a subset of the Cartesian product
A1  A2    An
All records should be members of the product
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19. Names  Marks  Marks  Marks  Marks
Tables (3)
Table T2: Course results
Name
Intro to Maths
Programming
Discrete
Maths
Computer
Systems
Cummings A B C
Jones D
Grant
Singh
French E
McKay
Cookson
Table is Cartesian product of
Names  Marks  Marks  Marks  Marks
Rows are elements
(Jones, B, C, B, D)
(Grant, C, B, A, C)
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20. Database operations DBMS provides operations on tables to
Extract information
Modify tables
Combine tables
Some of these operations are
Project (as in "to project")
Join
Select
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21. Project operation Selects columns of a table to form a new table
T3 = project(T1,{Name, Address}) yields
Name
Address
Jones
2 The Motte, Newton
Singh
4A New Road, Seaforth
Smith
17 The Crescent, Seaforth
21 Pudding Lane, Witham
Ching
Grant
18 Iffley Road, Reading
McKay
133 Uff Road, Reading
French
11 Finn Road, Newtown
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22. Join operation Takes two tables and joins their information
Puts together tuples which agree on attributes
join(T3, T2) yields
Name
Address
Intro to Maths
Programming
Discrete
Maths
Computer
Systems
Jones
2 The Motte, Newton B C D
Grant
18 Iffley Road, Reading A
Singh
4A New Road, Seaforth
French
11 Finn Road, Newtown E
McKay
133 Uff Road, Reading
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23. Select operation Extracts rows of a table which satisfy criteria
select(T1, (Sex = Male and Marital Status = Married))
ID No.
Name
Sex
DOB
Marital
Status
Address
Singh
Male
4.5.84
Married
4A New Road, Seaforth
Grant
9.7.83
18 Iffley Road, Reading
Contrast with
T5 = {(I,N,S,D,M,A) :
(I,N,S,D,M,A)  T1 and S = Male and M = Married}
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24. Summary You should now know: What equivalence relations are
Definition of set partition
Partial and total order of a relation
Concepts of database management systems
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25. Further reading
R. Haggarty. “Discrete Mathematics for Computing”. Pearson Education Ltd (Chapter 4)
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