Introduction
Relations
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics  Relations
132
Previous Lecture
Venn diagrams
Operations of
Connection to logic
Laws of set theory
square4
Intersection
square4
Union
square4
Symmetric difference
square4
Complement
square4
Difference
Discrete Mathematics  Relations
133
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
134
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 `1dimensional’ objects, then Cartesian
products are 2dimensional
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
135
Cartesian Product of More Than Two Sets
Instead of ordered pairs we may consider ordered
triples
,
or,
more general,
ktuples
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 AndreiBulatov
 Set Theory, Binary relation, Cartesian product, Ordered pair, Mark Jerry John Randy Aaron Ralph John Ralph Aaron Mark Jerry Randy

Click to edit the document details