Chapter
0
PRELIMINARIES
\
This
chapter
contains the
basic
material
on
sets,
functions,
relations
and
induction,
as
well
the
syater
of
real numbers.
Your
students
may
have
seen
a
good
share
this
material before.
You
will
need
to
make
some choices about
what
material
from
you cover
in
detail.
My
experience
has
been
that
most
students
at
thi~
level
have
little
appreci'ation
structure
system
real numbers
a complete ordered
field.
For
reason,
I
have
found
a
comprehensive
coverage
Section
0.5
is
very
important.
oes
for
Solutions
Exercises
3,
The
proof
is very similar to
that
(v)
Theorem
0.2.
4.
(i) of
Theorem
0.3.
6,
If
x
E
An
B, then
x
f
A
x E
particular,
x
E
A.
Hence,
A
fl
B C
x E
A,
then
x
E
A
U
B,
hence
A
C
A
U
B.
6.
x
E
C\B,
x
E
C
and
x
f
Since
A
C
B
x
j! B,
x
f!
Therefore,
x E
C\A.
Thus,
C\B
C
The converse is
false
as evidenced by
example
C
=
J,
A
=
-1,2,3)
and
B
=
(-3,2,3).
Here
C\A
=
but
A
$
k
and B
7.
A\(A\B)
=
B
if
only
B
C
To show
this, you
want
prove
A\(A\B)
=
A
0.
\
8,
Let
x
f
(A\B)
U
B\A).
Then
x f
A\l3
or
x
E
B\A.
x
E
A\B,
x
l
A
and
x
B,
hence,
x
E
A
U
B
and
I
x
A
P
3.
If
x
E B\A,
a similar
argument
shows
that
%
xEAU0
and
x
f
A
In
either
case,
x
E
(A
U
B
\
A
U
B
,
Now
assume
x f
U
B)\(A
0).
Then
x
E
hSB
and
xfAnB.
xEA,then
x$B
since xfAnB.
xEB,
xfA
since
x
f
A
B. But x
E
A
or
x
f
3
x
E
A\B
x
E
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(:\*{,
EL
u
(s\*)
=
AUB\AnB.
9.
point
Russell's
paradox
is to
show
student
using
a rule to define a
set can
lead
logical
difficulties.
However,
reassure
your
that
no such