Binary Relations
Closures
CITS2211 Discrete Structures
Lectures for Semester 2 2013
Part 4: Binary Relations
Unit coordinator: Mark Reynolds
Notes: Rachel Cardell-Oliver
Lecturer: Gordon Royle
September 3, 2013
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Binary Relations
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Cartesian product
Definition: The Cartesian product of sets S and T is a set of pairs:
S × T = {(s, t) | s ∈ S ∧ t ∈ T }
The Cartesian product can be generalised to n-tuples for more than
2 sets:
S1 × S2 × . . . × Sn = {(s1 , s2 , . . . sn ) | s1 ∈ S1 ∧ . . . ∧ sn ∈ Sn }
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Binary Relations
Definition: Given sets A and B, a binary relation R between A and
B is a subset of A × B.
The sets A and B are called the domain and codomain of R,
respectively.
We commonly use the notation a R b or a ∼R b or R(a, b) to
denote that (a, b) ∈ R
Relations are used in Computer Science to specify and reason
about data, for example, in data base theory and theories of
object-orientation.
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Defining Relations
Given that binary relations are sets, they can be defined by any of
the methods already given for defining sets:
Enumeration: {(0, 0), (1, −1), (2, −2)}
Abstraction: {(x, y ) | x ≤ y }
Relations can be combined using the set operations ∪ and ∩ and
complement.
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Types of Relations
Exercise: Express each of these characteristics using a predicate.
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Examples
The relation describing students and student numbers is
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Examples
The relation describing students and student numbers is
one-to-one
The relation describing students and computer accounts is
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Examples
The relation describing students and student numbers is
one-to-one
The relation describing students and computer accounts is
one-to-many
The relation describing students and degree programmes is
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Binary Relations
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Examples
The relation describing students and student numbers is
one-to-one
The relation describing students and computer accounts is
one-to-many
The relation describing students and degree programmes is
many-to-one
The relation describing students and unit enrolments is
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Binary Relations
Closures
Examples
The relation describing students and student numbers is
one-to-one
The relation describing students and computer accounts is
one-to-many
The relation describing students and degree programmes is
many-to-one
The relation describing students and unit enrolments is
many-to-many
Note that not all s ∈ S nor t ∈ T need appear in the relation.
∅ is a perfectly good relation!
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Properties of Relations
For any relation R : A → A
Definition: Reflexivity R is reflexive iff ∀x ∈ A. xRx
Example: Everyone likes themselves.
Definition: Symmetry R is symmetric iff ∀x, y ∈ A. xRy → yRx
Example: If x likes y, then y likes x.
Definition: Transitivity R is transitive iff
∀x, y , z ∈ A. xRy ∧ yRz → xRz
Example: If x likes y and y likes z, then x likes z too.
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Case study lecture
State the type of each of the following sets, stating any
assumptions you make. Which are relations?
likes, motherof , ancestorof , parentsof , connectedto
<, ≥, +, =, →
For each binary relation above, state whether it is reflexive,
transitive, symmetric. Give reasons to justify your answer.
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Definition: Antisymmetry R is antisymmetric iff
∀x, y ∈ A. xRy ∧ yRx → x = y
Example: If x likes y and y likes x then x is the same as y, that is
no pair of distinct people like each other
Definition: Asymmetry R is asymmetric iff ¬∃x, y ∈ A. xRy ∧ yRx
No one likes themselves and no pair of people like each other.
Definition: Irreflexivity R is irreflexive iff ¬∃x ∈ A. xRx
Example: No one likes themselves.
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Every relation R : A → B can be represented as a bipartite graph
G = (V , E ) by creating a left node for each element of A and a
right node for each element of B. We then create an edge between
a left node u and a right node v whenever uRv
Every relation on a single set A can be modelled as a directed
graph (albeit one that may contain loops).
Example: What relation is this? What properties does it have?
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Two important relations
Two of the most common operations on data are
1
ordering the data in some way
2
grouping the data in some way
Two important classes of relations that capture this are
(respectively)
1
partial orderings
2
equivalence relations (or partitions)
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Definition: A partial ordering on a set S is a binary relation R on
S that is reflexive, transitive, and antisymmetric.
Intuitively, the relation R orders pairs of elements in the set S,
although not necessarily every pair is ordered (hence partial).
Sometimes this definition is called a weak partial order.
Example: The relation ≤ on the integers Z is a partial order.
Example: The relation ⊆ on P(S) is a partial order on set S.
Exercise: Prove the above claims from the definitions.
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Definition: The pair (S, R) of relation R on set S is called a
partially-ordered set or poset.
Definition: A total order is a partial order in which every pair of
distinct elements is comparable.
Example: The relation ≤ on the reals R is a total order because for
any reals x and y either x ≤ y or y ≤ x.
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Definition: A Hasse diagram is a way to visualise a poset.
Conventions:
1
Every element of the relation set is shown as
a vertex in the graph
2
An edge is shown between each vertex and its
immediate successor vertices
3
Minimal element(s) at the bottom of the
page and max(s) at the top
4
Reflexive pairs, xRx are implied, not shown
5
Transitive pairs, xRz where xRy and yRz are
given, are implied, not shown
6
There is more than one way to lay out most
Hasse diagrams - finding a good layout can
be hard
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Examples for this Lecture
Find a piece of paper and draw diagrams (or list the pairs) for each
of the following relations on the sets R = {1, 2, 3, 4} and
S = {1, 2, 3, 4, 5, 6, 7, 8}
Example: x 6= y on R × R
Example: x = y + 1 on R × R
Example: x | y (that is x divides y, with no remainder) on S × S
Example: x =mod3 y (that is, x mod 3 = y mod 3) on S × S
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Questions
1
For each of your example relations, state whether it is
reflexive, transitive, symmetric, antisymmetric. Justify each
answer.
2
Make a note of any visual properties of the graph that will
help you remember these definitions.
(following the material from the remainder of this lecture)
3
Construct the reflexive and transitive closures of the first two
example relations on R × R
4
Construct a topological sort of the poset x | y
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Definition: An equivalence relation on a set S is a binary relation
r on S that is reflexive, transitive, and symmetric.
Intuitively, the relation r partitions elements in the set S.
Example: The relation = is an equivalence relation on set S.
Example: Congruence modulo n is an equivalence relation.
Example: Consider a relation “connected to” represented by
R : V → V on an undirected graph G = (V , E ) where uRv iff
there is a path in the graph from u to v . R is an equivalence
relation.
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Definition: Given an equivalence relation R : A → A, the
equivalence class of an element x ∈ A is the set of all elements of
A related to x by R. The equivalence class of x is denoted [x]
Definition: A partition of a set S is a set of non-empty,
pairwise-disjoint subsets of S whose union is S.
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Theorem: An equivalence relation on a set S uniquely determines a
partition of S, and a partition of S uniquely determines an
equivalence relation on S
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Partial orderings and Hasse diagrams are often used to represent
problems in task scheduling.
By convention the minimal task of the poset is shown on the left
and the maximal task on the right, so the task orders read from
left to right. Each task is associated with the time taken to
complete it: executionTime : tasks → N
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Definition: We can define a partial ordering on tasks, such that
xPy iff task x is a prerequisite for task y . P is a poset: it is
reflexive, antisymmetric and transitive. The Hasse diagram for the
poset (tasks, P) is called a PERT chart (program evaluation and
review technique chart).
Source http://www.yourdon.com/strucanalysis/
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Definition: the minimum time-to-completion for a process can
be calculated from a PERT chart:
The earliest start time (EST) for a task t is given by
EST (t) = max{ECT (u)|u is an immediate prereq. of t}
The earliest completion time (ECT) for a task t is given by
ECT (t) = EST (t) + executionTime(t)
The ECTs of tasks are calculated left-to-right, starting with
the minimal tasks
The minimum time-to-completion of a process is the latest
ECT of the tasks making up the process
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Definition: The critical path of a PERT chart is the sequence of
tasks that contribute to the minimum time-to-completion:
tasks on the critical path are called critical tasks
if any critical task is delayed, the minimum
time-to-completion of the process is increased
if any non-critical task is delayed, the critical path should be
re-computed.
Critical paths need not be unique.
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The process of learning to fly a plane can be broken down into the
following set of tasks.
Task
Days to perform Pre-requisite tasks
1. Pass medical test
1
None
2. Learn flight theory
7
1
3. Master plane controls
12
1
4. Learn engine maintenance
5
1
5. Pass initial flying test
1
3, 4
6. Pass theory exams
5
2
7. Learn navigation
7
5, 6
8. Learn ATC procedures
3
5
9. Pass final flying test
1
7, 8
Example: Construct a PERT chart for this process.
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Example: Use your PERT chart to determine the minimum
time-to-completion for the process (show working).
Example: Use your PERT chart to determine the critical path for
the process.
Example: Use your PERT chart to determine a topological sort for
the process.
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Definition: The transitive closure of a binary relation R on a set
A is the transitive relation R + on set A such that R ⊆ R + and R +
is minimal in that no smaller relation is a transitive closure of R.
Example: The reachability relation of a graph tells us which
points can be reached in any number of moves from each point.
The reachability relation is the transitive closure of the edge
relation E of a directed graph G = (V , E )
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Definition: The reflexive closure and symmetric closure are
defined similarly, as the smallest relation containing R that satisfies
the given property.
Example: There can be no antisymmetric closure of a relation
because antisymmetry is about the absence of pairs from a relation
(xRy and yRx must have x = y ) so adding pairs can’t make a
relation antisymmetric.
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Topological Sorting
A poset need not have every element comparable.
But given a poset (S, p), we can construct a new poset (S, q) by
adding tuples until it becomes a total order.
That is, a topological sort is a linear order that does not violate
the underlying partial order.
q is called a topological sort of p. q is not usually unique.
In computer science, topological sorting is used for
instruction scheduling in compiler optimisation,
cell evaluation order for spreadsheets,
determining compilation order in makefiles.
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Topological Sort in Practice
This process is easy to visualise from the Hasse diagram: the
topological sort gives a single path from min to max elements.
If p represents a PERT chart, then q defines a sequential order for
the tasks that respects the pre-requisite requirements
Topological sorts are not (usually) unique.
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Algorithm to Construct a Topological Sort
Input: A finite poset (S, p)
Output: A sequence of elements from S preserving the order p
A := S
while ( A 6= ∅ ) {
pick a minimal element b ∈ A;
A := A − {b};
output b
}
where a minimal element is one with no incoming edges:
¬∃ y ∈ S.(y , b) ∈ p
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A function is a binary relation R : A → B that can only associate
at most one element of B (namely, f (a)) with each element a ∈ A.
There will be more about functions in later units, but not in
cits2211.