1. CS1022 Computer Programming & Principles
Lecture 1
Relations

2. Plan of lecture Motivation Binary relations
Graphs to represent relations
Arrays to represent relations
Properties of relations
Closure of relations
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3. Why relations? Pieces of information are inherently related
People, addresses, age, what they have bought, etc.
When modelling information, we need to capture such relations:
Bob is the husband of Jane
Relation captured as an ordered pair (husband, wife),
An element of the Cartesian product of Male  Female sets
Another scenario: students registered for courses
Set S = {Bob, Tony, ...} of students
Set C = {CS1022, CS1015, ...} of courses
Cartesian product S  C = {(s, c) : s  S, c  C}
{(Bob,CS1022),(Bob,CS1015),(Bob, ...),(Tony,CS1022),(Tony,...)}
Very large set with all possible ways students can be registered
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4. What’s important in relations?
Binary relations model any relationship between elements of two sets
Important:
To know how to represent relations
Properties we want to study in some relations
Equivalence and partial orders
Databases borrow extensively from sets & relations
Whether you will work in IT or not, you will encounter information and how it is gathered, stored and managed
You might have to help choosing information to gather
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5. Binary relations
A binary relation between sets A and B is a subset R of the Cartesian product A  B, R  A  B
When A  B, we refer to R as a relation on A
Example: family tree
R = {(x, y) : x is a grandfather of y}
S = {(x, y) : x is a sister of y}
Fred & Mavis
Alice
Ken & Sue
John & Mary
Mike
Penny
Jane
Fiona
Alan
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6. Binary relations
A binary relation between sets A and B is a subset R of the Cartesian product A  B, R  A  B
When A  B, we refer to R as a relation on A
Example: family tree
R = {(x, y) : x is a grandfather of y}
R = {(Fred, Jane), (Fred, Fiona), (Fred, Alan), (John, Jane), (John, Fiona), (John, Alan)}
S = {(x, y) : x is a sister of y}
Fred & Mavis
Alice
Ken & Sue
John & Mary
Mike
Penny
Jane
Fiona
Alan
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7. (Jane, Fiona), (Jane, Alan), (Fiona, Jane), (Fiona, Alan)}
Binary relations
A binary relation between sets A and B is a subset R of the Cartesian product A  B, R  A  B
When A  B, we refer to R as a relation on A
Example: family tree
R = {(x, y) : x is a grandfather of y}
R = {(Fred, Jane), (Fred, Fiona), (Fred, Alan), (John, Jane), (John, Fiona), (John, Alan)}
S = {(x, y) : x is a sister of y}
S = {(Alice, Ken), (Sue, Mike), (Sue, Penny), (Penny, Sue), (Penny, Mike),
(Jane, Fiona), (Jane, Alan), (Fiona, Jane), (Fiona, Alan)}
Fred & Mavis
Alice
Ken & Sue
John & Mary
Mike
Penny
Jane
Fiona
Alan
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8. Graphs to represent relations (1)
Let A and B be two finite sets
Let R  A  B be a relation between A and B
We build a directed graph (or digraph):
For each (x, y)  R, draw an arrow (or arc) from x to y
Notice that arrow has a direction: from x to y
Example: Let A = {1, 3, 5, 7}, B = {2, 4, 6}
Relation V  A  B, V = {(x, y) : x  y}
Has the following graph 1 7 3 5 2 4 6
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9. Graphs to represent relations (2)
Example: Let A = {1, 2, 3, 4, 5, 6}
Relation R  A  A, R = {(x, y) : x is a divisor of y}
R = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 2), (2, 4), (2, 6), (3, 3), (3, 6), (4, 4), (5, 5), (6, 6)}
Graph:
divisor m of an integer n is an integer m that can be multiplied by some other integer o to produce n.
3 is a divisor of 6 since 3*2 = 6. 1 2 6 3 5 4
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10. Graphs to represent relations (3)
Graphs: very popular way to model information
Examples:
Cities and roads connecting them (some one-way)
“Friends” in social networks
Computing networks in general
N.B.: many algorithms (with guarantees) for graphs
Find out if two nodes are (not) connected
Find out if there are “cliques” (fully connected groups)
Find out “source” (a node from which all arrows leave) and “sinks” (nodes from which no arrows leave)
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11. Arrays to represent relations (1)
Arrays (a kind of table) to represent graphs
First, we
Label elements of sets A and B
List their elements in a particular order:
A = {a1i, a2j, , aqn} B = {b1k, b2l, , brm}
Notice:
We are not ordering sets – these are element positions
Whatever order, there will be an element 1, 2, etc.
Recall the discussion about subscripts to distinguish elements in a set and superscripts to order those elements.
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12. Arrays to represent relations (2)
Let A = {a1i, a2j, , aqn} B = {b1k, b2l, , brm}
Relation R  A  B: array M with q rows, r columns
Array M is called a q by r matrix
Rows labelled by elements of A (in chosen order)
Columns labelled by elements of B (in chosen order)
Entry in row i and column j is M(i, j) where:
M(i, j) = T (true) if, and only if, (ai, bj)  R
M(i, j) = F (false) if, and only if, (ai, bj)  R
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13. Arrays to represent relations (3)
Example: A = {a1, e3, c5, g7}, B = {b2, f4, d6}
Relation V  A  B, V = {(x, y) : x  y in the alphabet} 2 4 6 1 T 3 F 5 7 1 7 3 5 2 4 6
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14. Arrays to represent relations (4)
We could swap rows and columns around
We could use 0 and 1 instead of T and F
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15. Infix notation for relations
If R is a relation between two sets, we also write
x R y whenever (x, y)  R
x R y reads “x is R-related to y”
This is the “infix” notation to improve reading:
x is_a_sister_of y
x is_a_divisor_of y
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16. Notation for relations: summary
Relation R between two finite sets can be described:
In words (using a suitable predicate)
As a set of ordered pairs
As a digraph (directed graph)
As a matrix
Important: you are expected to
Describe relations in any representation
“Translate” between any two representations
Question: how would you represent relations using your “pet” programming language?
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17. Properties of relations (1)
We focus on relations on a single set A
A relation R on a set A (using infix notation) is:
Reflexive when x ((x  A)  (x R x))
Symmetric when xy ((x R y)  (y R x))
Antisymmetric when
xy (((x, y  A) and (x R y) and (y R x))  (x  y))
Transitive when
xyz ((x, y, z  A) and ((x R y) and (y R z)))  (x R z))
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18. Properties of relations (2)
A relation R on a set A is (using pairs) is
Reflexive when x ((x  A)  ((x, x)  R))
Symmetric when xy (((x, y)  R)  ((y, x)  R))
Antisymmetric when
xy (((x, y  A) and ((x, y)  R) and (x  y))  ((y, x)  R))
Transitive when
xyz (((x, y, z  A) and ((x, y)  R) and ((y, z)  R))
 ((x, z)  R))
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19. Properties of relations (3)
A relation R on a set A is (now using a graph) is
Reflexive if there is an arc from each vertex to itself
Symmetric if whenever there is an arc from x to y there is also an arc from y to x
Antisymmetric if whenever there is an arc from x to y and x  y, there is no arc from y to x
Transitive if whenever there is an arc from x to y and an arc from y to z, then there is also an arc from x to z
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20. Properties of relations (4)
Suppose relation x has the same age as y over People
It is reflexive as any x has the same age as itself
It is symmetric since x has the same age as y means the same as y has the same age as x
It is not antisymmetric because we can have many different people with the same age
It is transitive since if x has the same age as y and y has the same age as z, then x has the same age as z
"is taller than" has what properties?
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21. Properties of relations (5)
Antisymmetric?
has the same finger print as
has the same DNA as
has the same ID as
is parent of
is less than
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22. Properties of relations (6)
You should be able to
Check if a relation has any of the properties
Proof:
By a counter-example showing property does not hold
Assume premises (general) and reach conclusion, to show the property does hold
We might need to assume properties hold:
E.g., algorithm only works if relation is transitive
Algorithm needs pre-processing to check this is the case
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23. Closure of relations (1)
We can extend a relation R with more pairs with respect to a property or relations P
Example: what is the closure of the following ancestor with respect to transitivity?
xyz (((x, y, z  A) and ((x, y)  R) and ((y, z)  R))  ((x, z)  R))
Ancestor = {(bill, jill), (jill, kim), (phil,mary), (kim,phil), (mary,jane)}
Ancestor* = {(bill, jill), (jill, kim), (bill,kim), (phil,mary), (kim,phil), (jill,phil), (kim,mary), (bill, phil), (bill,mary), (mary,phil), (phil,jane), (bill,jane)}
Do we have all of them?? Do we have anything extra??
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24. Closure of relations (1)
We can extend a relation R with more pairs until a certain property P holds
If extended relation is smallest possible one, then
It is the closure of R with respect to property P
It is represented as R*
“Smallest possible”: only essential new pairs added
Formally, R* is the closure of R w.r.t. property P if:
R* has property P
R  R*
R* is a subset of any other relation that includes R and has property P
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25. Computing closure of relations
Algorithms exist to compute closures, e.g. transitive closure.
Algorithms will be quite specific to the property (e.g. transitive)
You need to search through all the ‘paths’ to find missing relations for transitive closure
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26. Summary You should now know: Why we need relations
How to represent relations
Properties of relations
Closure of relations
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27. Further reading
R. Haggarty. “Discrete Mathematics for Computing”. Pearson Education Ltd (Chapter 3)
Wikipedia’s entry
Wikibooks entry
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