The mathematics of relations
A set is a collection of objects, called its elements, or
members.
Examples:
the set of all dogs
the set of students in this class
the set comprising the numbers 1, 2, and 3
Typically, we use uppercase Roman letter A, B, ..., S, ... for
sets
For elements we typically use lowercase Roman letters a, b,
c, ..., x, y, z
‘aS’ means that a is a member of the set S
A set is standardly defined in two ways:
 by providing a list of its members, or
 by providing a property which picks out its members
in some unique way
Examples:
S1 = {1, 2, 3}
S2 = {x : x is a natural number between 0 and 4}
Beside these two ways, there is an informal way of defining
sets by drawing a diagram:
1
2
S3
3
Identity of sets (the principle of extension):
Two sets are identical iff they have the same members
Notation:
iff
stands for ‘if and only if’
therefore, S1 = S2 = S3
S1 = {1, 2, 3}
S2 = {x : x is a natural number between 0 and 4}
1
2
S3
3
From the principle of extension it follows:
{x, y} = {y, x}
Ordered pairs:
(x, y) or <x, y>
(x, y)  (y, x)
(x, y) = (x’, y’) iff both x = x’ and y = y’
Example:
{(x, y) : x is a natural number between 0 and 4, y = x2} =
= {(1, 1), (2, 4), (3, 9)}
Special symbol for the empty set: 
Quantifiers: , 
universal quantifier: x(x  1)
existential quantifier: y(y2 = 1)
Subsets:
S  S’ (S is a subset of S’) iff x(if xS, then xS’)
Real subsets:
S  S’ (S is a real subset of S’) iff S  S’ but not S’  S
All of known mathematics can be built out of sets and
logic.
In particular, everything in this course can.
Relations
In math, science, and everyday life we are continually
concerned with relationships among objects.
It is therefore useful to have a mathematical language that
can talk about relations.
Examples of relations:
x is bigger than y
x is better than y
x loves y
x causes y
x is preferred to y
Notation:
 We denote relations by uppercase Roman letters, e.g.
R, R’.
 Some important relations have special symbols, e.g. >,
=.
 We write xRy to mean that relation R holds between x
and y
Defining relations:
We will use three ways to define particular relations:
1. List all the objects of interest between which the
relation holds
Examples:
Let S = {Victoria, Alberta, Manitoba, Edmonton,
Winnipeg, B.C.} be our set of objects of interest.
Let R be the relation ‘is capital of’
Then R is defined as:
Victoria R B.C.
Edmonton R Alberta
Winnipeg R Manitoba
Let S’ = {1, 2, 3}
Then the relation ‘>’ is defined on the set S’ as:
3>2
3>1
2>1
2. By description
xRy holds iff x is the capital of y and both x and and y are
in S
x > y iff x is greater than y and both x and y are in S’
3. By picture
Victoria
B.C.
Edmonton
Alberta
Winnipeg
Manitoba
Formal definition of relations:
Relations are sets
A binary relation is a set of ordered pairs
A relation is always defined on a set
Thus R is defined on S, > is defined on S’
R = {(Victoria, B.C.), (Edmonton, Alberta), (Winnipeg,
Manitoba)}
> = {(3, 2), (3, 1), (2, 1)}
Properties of relations:
A relation R on a set S is:
reflexive
iff
symmetric
iff
transitive
iff
total (connected)
(xS) xRx
(x,yS) if xRy then yRx
(x,y,zS) if xRy and yRz, then xRz
iff (x,yS) either xRy or yRx
Functions
Functions are relations
Definition:
A relation F on a set S is a function if F relates all elements
of S to at most one element of S, i.e.
(x,y,zS) if xFy and xFz, then y  z
Notation:
 Functions are denoted by the lowercase Roman letters:
f, g, u, p
 If f is a function, we typically write f(x)  y instead of
xfy
 If f(x)  y, then x is called the argument of function f,
and y is called the value of f for x.
Examples:
‘son-of-father’
‘is-capital-of’
‘is-parent-of’
is a function
is a function
is not a function
Preference relations and score functions
Decision theory studies relations of preference among a
given set of options.
Notation:
Let O be a set of options among which an agent A is
choosing (flavours of ice cream, stocks, lotteries, strategies
in a game, etc.).
1. We write x y to mean that ‘A prefers option x to
option y’
2. We write x ~ y to mean that ‘A is indifferent between
x and y’
3. We write x y to mean that ‘A does not prefer option
y to option x’
 is called strict preference relation
 is weak preference relation
~ is indifference relation
Some preference relations are rankings of the available
options. Rankings of options can be represented by
assigning a number to each option such that an option is
preferred just in case its number is greater.
Example:
Suppose Jane ranks TV shows as follows. Seinfeld is the
best, then Friends, then the Simpsons
score(Seinfeld)  3
score(Friends)  2
score(Simpsons)  1
x y iff score(x) > score(y)
It’s only order that matters:
score2(Seinfeld)  1000
score2(Friends)  2
score2(Simpsons)  –123
x y iff score2(x) > score2(y)
We take the weak preference relation  as a primitive
notion
Strict preference and indifference relations are then defined
as:
xy
x~y
iff
iff
x  y and not y  x
x  y and y  x
Definition of the score function (or utility function):
A score function or utility function u assigns a number to
each option in a given set O. A utility function represents a
preference relation  on a set of options O just in case
(x,yO) x  y holds iff u(x)  u(y)
Rational preferences
Definition:
A preference relation is rational iff it is:
1. total
2. reflexive
3. transitive
Theorem 1:
If a weak preference relation  is rational, then
1. the corresponding strict preference relation  is
transitive
2. the corresponding indifference relation ~ is reflexive
and transitive
Theorem 2
Given a finite set of options, there is a score function that
represents a weak preference relation  just in case  is
rational.
Money pumping
coke
3
sprite
2
juice
1
1
2
csjc
3
1
2
3