BASIC CONCEPTS OF SET THEORY

SETS
The concept of a set and the properties of sets derived from certain fundamental
assumptions or axioms form the basis of all modern mathematics.
Axiomatic Set Theory
derives the properties
ot
sets but does not define a set itself.
The theory is based on an
intuitive understanding of a set as a collection of entities or elements
Notation: The standard notation for a set is as follows: The elements of the set under
consideration are separated by commas and enclosed in curly brackets.
For example a set
X with elements
Xl>
X2,
X3
and
X4
is denoted as X=
{Xl,
x2, Xj,
X4}
Sets and their
elements and defined terms are denoted in italics to differentiate them from other entities.
Properties of Sets
a. Set Membership
Sets consist of
elements
or
members.
The elements of a set are its fundamental or atomic
units
A set may also be an element of another
set For example, a line is set of points;
the set of all lines defining a plane is a set of sets (ofpoints)
All possible nesting of sets
of sets are allowed as elements of other sets
Notation: If
X
is an element of a set X,
X
is said to be contained in X and is denoted as
follows:
x
E
X
b. Set Equality
Two sets X and
Y
are said to equal if and only if they have the same elements. The
equality of two sets, say X and
Y
is denoted as X
=
Yand their inequality by X
=
Y
c. Subsets
Most of set theory is aimed at creating new sets out of old ones.
If
X and Yare sets such
that every element of X is also an element of
Y,
then X is said to be a
subset
of
Yor
that
Y
includes
X
This fact is denoted as follows: X
c
YorY
:::>
X
Example:
Let
Y
be a set of say a hundred individuals.
we consider only those that are married and
call the subcollection X, then X
c
Y
The set X itself is denoted as: X
=
{x
E
Y
.
x is
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j,
where
x
is any element
of
Y
The part enclosed by the brackets is to be read as
"x
in
Y
such that
x
is married"
Some Important Properties of Subsets:
i).
According to the defintion
of a subset any set
X
should include itself, i.e.
X
c
X
This property
of set inclusion is said to be
reflexive.
ii.)
If X,
Y
and
Z
are sets such that
X
c
Y,Y
c
Z
then
X
c
Z .
In this case set inclusion
is said to be
transitive
iii).
If
Y
c
X
and
X
c
Y
then
X
and
Y
have the same elements so that
X =
Y
In fact the
commonly used technique to prove the equality of two sets is to show that they include
each other
Note: i)
The properties
of belonging (
E
)
and inclusion (
c)
are conceptually very
different
Inclusion is always reflexive whereas belonging is
not
It
is always true that a
set X
c X
However, it would be difficult, if at all, to come up with an example
of a set
XEX
ii)
A set
X
is called
countable
if there exists a onetoone correspondence between the
set
of all integers and the elements of
X
A set
X
isfinite
if for some positive integer
'n'
there exists a onetoone correspondence between the elements of the set
(I, 2, 3,.
..n)
and the elements of
X. For example, the rational numbers are countable, whereas the set
of reals are uncountable
d. Power Set:
Theorem:
If
X =
[x
j'
x
z,
x,J
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 Fall '05
 JUNG
 Set Theory, Sets, Probability, AiB

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