Transcript Mathematics for Computing - Birkbeck, University of London

Mathematics for Computing
Lecture 7:
Relations
Dr Andrew Purkiss-Trew
Cancer Research UK
e-mail: [email protected]
Material to be covered today
Relations
Definition of a relation
Representation of relations
Equivalence relations
Partial order relations
Partitioning
Cartesian Products (from last time)
A x B = {(x,y): xA and yB}
Example: A = {1,3,5}, B = {2,4}
A x B = {(1,2),(1,4),(3,2),(3,4),(5,2),(5,4)}
Relations
If A and B are sets then
A binary relation R, from A to B assigns to
each ordered pair (a,b) in A x B only (and
only one) of the following statements:
i) “a is related to b”, written a R b
ii) “a is not related to b”, written a R b
Relations 2
A relation R from A to B is a subset (R*) of
AxB
R* = {(a,b): a is related to b} = {(a,b): a R b}
or any subset, R*, of A x B, uniquely defines a
relation R from A to B
a R b whenever (a,b)  R*
Relation of set on itself
Relation on a set A is a subset of A x A.
e.g. A = {1,2,3,4}, relation ‘greater than’
{(2,1),(3,1),(3,2),(4,1),(4,2),(4,3)}
Graphical Representation
Directed Graph
1
2
3
4
Matrix Representation
1
2
3
4
1
F
T
T
T
2
F
F
T
T
3 4
F F
F F
F F
T F
Relation Definitions
If R is a relation on a set A then:
 R is reflexive if x R x for all x  A
 R is irreflexive if there are no elements of A for
which x R x
 R is symmetric if x R y implies y R x, for all x, y
A
 R is antisymmetric if x R y and y R x imply x = y,
for all x, y  A
 R is transitive if x R y and y R z imply x R z, for
all x, y, z  A
Equivalence Relation
An equivalence relation is one that is:
 reflexive
 symmetric and
 transitive
Partial Order Relation
An partial order relation is one that is:
 reflexive
 antisymmetric and
 transitive
Partitioning and equivalence
If A is a set, then a partition of A is a set of subsets
such that every element in A is an element of
exactly one of the subsets.
If A is a set and R is an equivalence relation on A.
For each element x of A, let E(x) be the subset of
A defined: E(x)={yA: y R x}. The set of all E(x)
is a partition of A.
The subsets E(x) are the equivalence classes of R