Introduction
Relations
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics - Relations
13-2
Previous Lecture
Venn diagrams
Operations of
Connection to logic
Laws of set theory
s
Intersection
s
Union
s
Symmetric difference
s
Complement
s
Difference
Discrete Mathematics - Relations
13-3
Relations
`
Relation
’,
the connection between things or people
Between people, family relations
`to be brothers’
x
is a brother of
y
`to be older’
x
is older than
y
`to be parents’
x
and
y
are parents of
z
Between things, numerical relations
`to be greater than’
x < y
on the set of real numbers
`to be divisible by’
x
is divisible by
y
on the set of integers
Between things and people, legal relations
`to be an owner’
x
is an owner of
y
Discrete Mathematics - Relations
13-4
Cartesian Product
The
Cartesian product
of sets
A
and
B,
denoted by
A
×
B,
is
the set of all
ordered pairs
of elements from
A
and
B.
A
×
B = { (a,b) | a
∈
A,
b
∈
B }
The elements of the Cartesian product are ordered pairs.
In
particular,
(a,b) = (c,d)
if and only if
a = c
and
b = d.
If sets are thought of as `1-dimensional’ objects, then Cartesian
products are 2-dimensional
1
2
3
4
5
1
2
3
{1,2,3,4,5}
×
{1,2,3}
3
1
5
2
(2,5)
×
(1,3)
1
2
3
4
5
6
7
8
a
b
c
d
e
f
g
h
Discrete Mathematics - Relations
13-5
Cartesian Product of More Than Two Sets
Instead of ordered pairs we may consider ordered
triples
,
or,
more general,
k-tuples
.
(a,b,c),