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Basic Concepts of the Theory of Sets
In
discussing
any
branch
of
mathematics,
be
it
analysis,
algebra,
or
geometry,
i
t
is
helpful
to
use
the
notation
and
terminology
of
set
theory.
This
subject,
wh
ich
was
developed
by
Boole
and
Cantor
in
the
latter
part
of
the
19
th
century,
has
had
a
profound
influence
on
the
development
of
mathematics
in
the
20
th
c
entury.
It
has
unified
many
seemingly
disconnected
ideas
and
has
helped
to
re
duce
many
mathematical
concepts
to
their
logical
foundations
in
an
elegant
and
systematic
way.
A
thorough
treatment
of
theory
of
sets
would
require
a
lengt
hy
discussion
which
we
regard
as
outside
the
scope
of
this
book.
Fortunately,
the
basic
noticns
are
few
in
number,
and
it
is
possible
to
develop
a
working
knowledge
of
the
methods
and
ideas
of
set
theory
through
an
informal
discussi
on
.
Actually,
we
shall
discuss
not
so
much
a
new
theory
as
an
agreement
ab
out
the
precise
terminology
that
we
wish
to
apply
to
more
or
less
familiar
ide
as.
In
mathematics,
the
word
“
set
”
is
used
to
represent
a
collection
of
objects
vi
ewed
as
a
single
entity
The
collections
called
to
mind
by
such
nouns
as
“
flock
”
,
“
tribe
”
,
‘
crowd
”
,
“
team
’
,
are
all
examples
of
sets,
The
individual
objects
in
the
collection
ar
e
called
elements
or
members
of
the
set,
and
they
are
said
to
belong
to
or
to
be
contained
in
the
set.
The
set
in
turn
,is
said
to
contain
or
be
composed
o
f
its
elements.
We
shall
be
interested
primarily
in
sets
of
mathematical
objects:
sets
of
numb
ers,
sets
of
curves,
sets
of
geometric
figures,
and
so
on.
In
many
applications
it
is
convenient
to
deal
with
sets
in
which
nothing
special
is
assumed
about
th
e
nature
of
the
individual
objects
in
the
collection.
These
are
called
abstract
se
ts.
Abstract
set
theory
has
been
developed
to
deal
with
such
collections
of
arbi
trary
objects,
and
from
this
generality
the
theory
derives
its
power.
NOTATIONS.
Sets
usually
are
denoted
by
capital
letters:
A,B,C,
?
.X,Y,Z
;
el
ements
are
designated
by
lower-case
letters:
a,
b,
c,
?
.x,
y,
z.
We
use
the
spe
cial
notation
X
∈
S
To
mean
that
“
x
is
an
element
of
S
“
or
”
x
belongs
to
S
”
.
If
x
does
not
belong
to
S,
we
write
x
∈
S.
When
convenient
,we
shall
designate
sets
by
disp
laying
the
elements
in
braces;
for
example
，
the
set
of
positive
even
integers
l
ess
than
10
is
denoted
by
the
symbol{2,4,6,8}whereas
the
set
of
all
positive
e
ven
integers
is
displayed
as
{2,4,6,
?
},the
dots
taking
the
place
of
“
and
so
o
n
”
.
The
first
basic
concept
that
relates
one
set
to
another
is
equality
of
sets:
DEFINITION
OF
SET
EQUALITY
Two
sets
A
and
B
are
said
to
be
equal
(or
identical)if
they
consist
of
exactly
the
same
elements,
in
which
case
we
wr
ite
A=B.
If
one
of
the
sets
contains
an
element
not
in
the
other
,we
say
the
s
ets
are
unequal
and
we
write
A
≠
B.
SUBSETS.
From
a
given
set
S
we
may
form
new
sets,
called
subsets
of
S.
For
example,
the
set
consisting
of
those
positive
integers
less
than
10
which
a
re
divisible
by
4(the
set{4,8})is
a
subset
of
the
set
of
all
even
integers
less
th
an
general,
we
have
the
following
definition.
DEFINITION
OF
A
SUBSET.
A
set
A
is
said
to
be
a
subset
of
a
set
B,
and
we
write
A
B
Whenever
every
element
of
A
also
belongs
to
B.
We
also
say
that
A
is
conta
ined
in
B
or
B
contains
A.
The
relation
is
referred
to
as
set
inclusion.
The
statement
A
B
does
not
rule
out
the
possibility
that
B
A.
In
fact,
we
m
ay
have
both
A
B
and
B
A,
but
this
happens
only
if
A
and
B
have
the
same
elements.
In
other
words,
A=B
if
and
only
if
A
B
and
B
A
.
This
theorem
is
an
immediate
consequence
of
the
foregoing
definitions
of
equ
ality
and
inclusion.
If
A
B
but
A
≠
B,
then
we
say
that
A
is
a
proper
subset
of
B:
we
indicate
this
by
writing
A
B.
In
all
our
applications
of
set
theory,
we
have
a
fixed
set
S
given
in
advance,
and
we
are
concerned
only
with
subsets
of
this
given
set.
The
underlying
set
S
may
vary
from
one
application
to
another;
it
will
be
referred
to
as
the
uni
versal
set
of
each
particular
discourse.
The
notation
{X
∣
X
∈<
/p>
S.
and
X
satisfies
P}
will
designate
the
set
of
all
elements
X
in
S
which
satisfy
the
property
P.
Wh
en
the
universal
set
to
which
we
are
referring
id
understood,
we
omit
the
refe
rence
to
S
and
we
simply
write{X
∣
X
satisfies
P}.This
is
read
“
the
set
of
all
x
such
that
x
satisfies
p.
”
Sets
designated
in
this
way
are
said
to
be
describ
ed
by
a
defining
property
For
example,
the
set
of
all
positive
real
numbers
co
uld
be
designated
as
{X
∣
X>0};the
universal
set
S
in
this
case
is
understood
t
o
be
the
set
of
all
real
numbers.
Of
course,
the
letter
x
is
a
dummy
and
may
be
replaced
by
any
other
convenient
symbol.
Thus
we
may
write
{x
∣
x>0}={y
∣<
/p>
y>0}={t
∣
t>0}
and
so
on
.
It
is
possible
for
a
set
to
contain
no
elements
whatever.
This
set
is
called
th
e
empty
set
or
the
void
set,
and
will
be
denoted
by
the
symbol
φ
.We
will
co
nsider
φ
to
be
a
subset
of
every
set.
Some
people
find
it
helpful
to
think
of
a
set
as
analogous
to
a
container(such
as
a
bag
or
a
box)containing
certain
obje
cts,
its
elements.
The
empty
set
is
then
analogous
to
an
empty
container.
To
avoid
logical
difficulties,
we
must
distinguish
between
the
element
x
and
t
he
set
{x}
whose
only
element
is
x
,(A
box
with
a
hat
in
it
is
conceptually
d
istinct
from
the
hat
itself.)In
particular,
the
empty
set
φ
is
not
the
same
as
the
set
{
φ
}.In
fact,
the
empty
set
φ
contains
no
elements
whereas
the
set
{
φ
}
h
as
one
element
φ
(A
box
which
contains
an
empty
box
is
not
empty).Sets
con
sisting
of
exactly
one
element
are
sometimes
called
one-element
sets.
UNIONS,INTERSECTIONS
,
COMPLEMENTS.
From
two
given
sets
A
and
B,
we
can
form
a
new
set
called
the
union
of
A
and
B.
This
new
set
is
deno
ted
by
the
symbol
A
∪
B(read:
“
A
union
B
”
)
And
is
defined
as
the
set
of
those
elements
which
are
in
A,
in
B,
or
in
both.
That
is
to
say,
A
∪
B
is
the
set
of
all
elements
which
belong
to
at
least
one
of
the
sets
A,B.
Similarly,
the
intersection
of
A
and
B,
denoted
by
A
∩
B(read:
“
A
intersection
B
”
)
Is
defined
as
the
set
of
those
elements
common
to
both
A
and
B.
Two
sets
A
and
B
are
said
to
be
disjoint
if
A
∩
B
=
φ
.
If
A
and
B
are
sets,
the
difference
A-B
(also
called
the
complement
of
B
rel
ative
to
A)is
defined
to
be
the
set
of
all
elements
of
A
which
are
not
in
B.
Thus, by
definition,
A-
B={X|X
∈
A
and
X
B}
The
operations
of
union
and
intersection
have
many
formal
similarities
with
(as
well
as
differences
from)
ordinary
addition
and
multiplications
of
union
and
intersection,
it
follows
that <
/p>
A
∪
B=B
∪<
/p>
A
and
A
∩
B=B
∩
A.
That
is
to
say,
uni
on
and
intersection
are
commutative
operations.
The
definitions
are
also
phrase
d
in
such
a
way
that
the
operations
are
associative:
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