Introduction
Sets
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics - Sets
11-2
What is a Set
A
set
is an unordered collection of objects
This is not a rigorous definition!!!
Every `conventional’ definition reduces the defined concept to a
wider, more general, concept.
For example,
`A
cow
is a big animal with horns and four legs in
the corners’
There is no more general concept than sets.
Therefore a rigorous
definition is impossible.
If we use the `definition’ above,
we get
`naïve set theory’
Otherwise we need
axiomatic set theory
,
Zermelo-Frenkel axioms
(introduced by Skolem)
ZF or ZFC.
Discrete Mathematics - Sets
11-3
Elements, Describing a Set
The objects in a set are called
elements
or
members
of the
set
One way to describe a set is to list its elements
a
∈
A
– a
is an element of
A,
a
belongs to
A
a
∉
A
– a
is not an element of A,
a
does not belong to
A
{0,1,2,3,4,5,6,7,8,9}
– the set of digits
{a,b,c,…,x,y,z}
– set of Latin letters, alphabet
A set can be an element of another set
Set of alphabets:
{{a,b,c,…}, {
α
,
β
,
γ
,…},…}
Discrete Mathematics - Sets
11-4
Set Builder
Big sets can be described using
set builder
:
{x | P(x)},
the set of all
x such that
P(x)
{x | there is
y
such that
x = 2y},
the set of even numbers
=
{x |
5
y (x =2y)}
{x | x
is a black cow}
N
= {0,1,2,3,…},
the set of natural numbers
Z
= {…,-2,-1,0,1,2,3,…},
the set of integers