Chapter
0
PRELIMINARIES
\
This
chapter
contains the
basic
material
on
sets,
functions,
relations
and
induction,
as
well
as
the
syater
of
real
numbers.
Your
students
may
have
seen
a
good
share
of
this
material
before.
You
will
need
to
make
some
choices
about
what
material
from
this
chapter
you
cover
in
detail.
My
experience
has
been
that
most
students
at
t h i ~
level
have
little
appreci'ation
of
the
structure
of
the
system
of
real
numbers
as
a
complete
ordered
f i e l d .
For
that
reason,
I
have
found
that
a
comprehensive
coverage
of
Section
0.5
is
very
important.
oes
for
Solutions
to
Exercises
3,
The
proof
is
very
similar
to
that
of
(v)
of
Theorem
0.2.
4.
The
proof
is
very
similar
to
that
of
(i)
of
Theorem
0.3.
6,
If
x
E
A
n
B,
then
x
f
A
and
x
E
3 ,
in
particular,
x
E
A .
Hence,
A
fl
B
C
A .
If
x
E
A ,
then
x
E
A
U
B,
hence
A
C
A
U
B.
6 .
If
x
E
C\B,
then
x
E
C
and
x
f
B.
Since
A
C
B
and
x
j!
B ,
then
x
f!
A .
Therefore,
x
E
C\A.
Thus,
C\B
C
C\A.
The
converse
is
false
as
evidenced
by
the
example
C
=
J ,
A
=
1,2,3)
and
B
=
(3,2,3).
Here
C\A
=
C\B
but
A
$
k
and
B
A .
7.
A\(A\B)
=
B
if
and
only
i f
B
C
A .
To
show
this,
you
may
want
to
prove
that
A\(A\B)
=
A
fl
0.
\
8,
L e t
x
f
(A\B)
U
B \ A ) .
Then
x
f
A\l3
or
x
E
B\A.
If
x
E
A \ B ,
then
x
l
A
and
x
f
!
B ,
hence,
x
E
A
U
B
and
I
x
f
!
A
P
3.
If
x
E
B\A,
a
similar
argument
shows
that
%
x E A U 0
and
x
f
A
fl
B.
In
either
case,
x
E
(A
U
B
\
A
U
B
,
Now
assume
x
f
(A
U
B)\(A
fl
0).
Then
x
E
h
S
B
and
x
f
A
n
B
.
If
x E A , t h e n
x $ B
since
x f A n B .
If
xEB,
then
x f A
since
x
f
A
fl
B.
But
x
E
A
or
x
f
3
hence,
x
E
A\B
or
x
E
B\A.
Thus,
x
E
$A\B)
U
$ B \ A ~ .
We
have
shown
that
A\B)
U
B\A)
C
A
U
j*
v
sji[*
n
sj
$
(r\a]'SA(:\*{,
E L
u
(s\*)
=
A
U
B
\
A
n
B
.
9 .
The
point
of
Russell's
paradox
is
to
show
the
student
that
using
a
rule
to
define
a
set
can
lead
t o
logical
d i f f i c u l t i e s .
However,
reassure
your
students
that
no
such
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 Spring '09
 kuiper
 Real Numbers, Sets, Trigraph, hat, ince

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