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§3.DEFINITION
Discrete
Mathema6cs
is
the
Math
needed
in
decision‐making
in
non‐
con6nuous
situa6ons.
Thus,
it
mainly
deals
with
discrete
objects,
their
best
examples
being
the
finite
sets.
However,
we
will
have
to
consider
infinite
sets
as
well.
In
broad
terms,
think
of
discrete
objects
as
objects
that
can
be
separated
from
one
another.
The
material
we
will
discuss
differs
fundamentally
from
the
topics
of
a
typical
Calculus
course,
as
the
realm
of
Calculus
is
the
conBnuous.
J.
Viola‐Prioli
It
can
be
said
that
such
a
course
can
be
thought
of
as
a
dramaBc
producBon.
But
then,
who
are
the
characters?
The
three
main
characters
are:
DefiniBon,
Theorem
and
Proof.
(there
is
also
an
"evil"
character,
to
be
menBoned
at
a
later
stage)
A
mathema6cal
defini6on
must
be
absolutely
precise
(whether
it
is
an
object,
a
concept,
a
command,
etc)
with
no
room
for
ambiguity.
DefiniBons
are
not
to
be
argued
about:
they
are
accepted
as
long
as
they
do
not
contradict
what
is
known
to
be
true.
In
Mathema6cs
is
crucial
to
understand
a
defini6on
before
proceeding.
In
what
follows
we
assume
knowledge
of
the
set
of
integers
(denoted
by
Z)
and
the
three
basic
operaBons:
+,
−,
×
and
the
order
relaBons
<,
≤
,
>,
≥
The
natural
numbers
(denoted
by
N)
are
the
non‐negaBve
integers:
therefore,
N=
{0,
1,
2,
3,
.
.
.}
whereas
Z=
{
.
.
.
.
,−2,
−1,
0,
1,
2,
3,
.
.
.}
J.
Viola‐Prioli
NOTE:
some
authors
define
N
as
the
set
of
posiBve
integers,
thus
deleBng
0
from
N.
However,
we
will
adhere
to
our
textbook's
definiBon.
Some
notaBons
and
remarks
are
worth
considering:
The
symbol
∈
means
"belongs
to".
Thus,
we
can
write
4
∈
N.
Their
negaBon
is
denoted
by
∉
.
We
use
curly
brackets
{
}
to
indicate
a
set
of
elements.
For
simplicity
we
someBmes
write
p.q
(or
pq)
to
denote
the
product
p×q
FracBons
(like
3/7)
play
almost
no
role
in
this
course,
as
we
deal
with
integer
numbers
almost
exclusively.
Next
we
will
explore
three
important
concepts.
J.
Viola‐Prioli
DEFINITION.
Given
integers
a
and
b
we
say
that
a
is
divisible
by
b
if
there
exists
an
integer
c
so
that
a
=
b×
c.
There
are
equivalent
ways
to
express
that
a
is
divisible
by
b:
a
is
a
mulBple
of
b
b
divides
a
b
is
a
factor
of
a
b
is
a
divisor
of
a
In
any
case
a
verBcal
bar
is
the
symbol
to
be
used:
hence,
b|a
is
read
"b
divides
a"
or
any
of
the
equivalent
forms.
CAUTION:
3|6
is
true
(because
6
=
2×3
is
a
mulBple
of
3).
Observe
that
3|6
is
a
sentence,
is
not
a
number!
On
the
other
hand,
3/6
is
a
fracBon
(whose
value
is
.5)
and
so
is
6/3,
which
equals
2.
Therefore
"6|48"
is
true,
"4|10"
is
false,
"1|a
for
every
integer
a"
is
true,
“−3|12"
is
true
(since
12
=
(−3)×(−4),
so
here
−4
=
c
of
our
definiBon
above).
Also
observe
that
a|a
for
every
integer
a,
including
a
=
0.
J.
Viola‐Prioli
DEFINITION.
An
integer
a
is
even
if
it
is
a
mulBple
of
2.
Therefore,
a=2c
for
some
integer
c.
The
even
numbers
are
thus
{
.
.
.
,
−4,
−2,
0,
2,
4,
6,
8,
.
.
.}.
Observe
that
0
is
even,
as
0
=
2
×
0.
DEFINITION.
An
integer
a
is
odd
if
a
=
2b
+
1,
for
some
integer
b.
The
odd
numbers
are
therefore
{
.
.
.
,
−3
,
−1,
1,
3,
5,
7,
.
.
.}.
For
instance,
‐7
is
odd
because
we
can
write
−7
=
2(−4)
+
1.
NoBce
that
a
number
is
odd
if
it
comes
immediately
ajer
an
even
number.
DEFINITION.
A
posiBve
integer
p
is
called
a
prime
if
p>1
and
its
only
posiBve
divisors
are
1
and
p
itself.
So
for
p
to
be
a
prime
number
it
must
saBsfy
these
three
condiBons
simultaneously:
1)
p
must
be
an
integer
2)
p
must
be
greater
than
1
3)
the
only
posiBve
divisors
of
p
must
be
1
and
p.
J.
Viola‐Prioli
5
Examples
of
numbers
that
are
not
primes:
(1
fails)
,
−7
(2
fails),
18/7
(1
fails),
14
(3
fails
as
14
=
2
×
7)
The
first
prime
numbers
are
2,
3,
5,
7,
11,
13,
17,
19,
23,
29,
.
.
.
.
€
We
observe
that
the
prime
numbers,
other
than
2,
are
odd.
However,
15
is
odd
but
is
not
a
prime
number.
In
a
later
chapter
we
will
prove
that
there
are
infinitely
many
prime
numbers.
DEFINITION.
A
posiBve
integer
a
is
called
composite
if
there
exists
an
integer
b
such
that
1
<
b
<
a
and
b|a.
In
other
words,
a
admits
a
proper
divisor.
Recall
that
every
number
is
divisible
by
itself
and
by
1:
what
makes
a
number
composite
is
the
fact
that
admits
other
divisors.
NoBce
that
composite
numbers
"break
down
as
a
product
of
strictly
smaller
natural
numbers":
if
a
is
composite,
according
to
the
definiBon
we
have
a
proper
factor
b.
Hence
a
=
b
c
and
since
1
<
b
<
a
it
is
clear
that
1
<
c
<
a.
Thus,
a
is
the
product
of
two
smaller
factors.
J.
Viola‐Prioli
Observe
that
according
to
the
definiBons
of
this
secBon,
0
and
1
are
special
in
that
they
are
neither
prime
nor
composite.
EXERCISE
a)
Find
three
different
prime
numbers
p,
q
and
z
such
that
pqz
+
1
is
prime.
b)
Find
three
different
prime
numbers
a,
b
and
c
such
that
abc
+
1
is
composite
J.
Viola‐Prioli