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What’s It All About?
•
Continuous mathematics—
calculus
—considers objects
that vary continuously
◦
distance from the wall
•
Discrete mathematics considers
discrete
objects, that
come in
discrete
bundles
◦
number of babies: can’t have 1.2
The mathematical techniques for discrete mathematics
diFer from those for continuous mathematics:
•
counting/combinatorics
•
number theory
•
probability
•
logic
We’ll be studying these techniques in this course.
1
Why is it computer science?
This is basically a mathematics course:
•
no programming
•
lots of theorems to prove
So why is it computer science?
Discrete mathematics is the mathematics underlying al
most all of computer science:
•
Designing highspeed networks
•
±inding good algorithms for sorting
•
Doing good web searches
•
Analysis of algorithms
•
Proving algorithms correct
2
This Course
We will be focusing on:
•
Tools for discrete mathematics:
◦
computational number theory (handouts)
*
the mathematics behind the RSA cryptosystems
◦
a little graph theory (Chapter 3)
◦
counting/combinatorics (Chapter 4)
◦
probability (Chapter 6)
*
randomized algorithms for primality testing, rout
ing
◦
logic (Chapter 7)
*
how do you
prove
a program is correct
•
Tools for proving things:
◦
induction (Chapter 2)
◦
(to a lesser extent) recursion
±irst, some background you’ll need but may not have .
. .
3
Sets
You need to be comfortable with set notation:
S
=
{
m

2
≤
m
≤
100
, m
is an integer
}
S
is
the set of
all
m
such that
m
is between 2 and 100
and
m
is an integer.
4
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View Full DocumentImportant Sets
(More notation you need to know and love .
. . )
•
N
(occasionally
IN
): the nonnegative integers
{
0
,
1
,
2
,
3
, . . .
}
•
N
+
: the positive integers
{
1
,
2
,
3
, . . .
}
•
Z
: all integers
{
. . . ,

3
,

2
,

1
,
0
,
1
,
2
,
3
, . . .
}
•
Q
: the rational numbers
{
a/b
:
a, b
∈
Z, b
6
= 0
}
•
R: the real numbers
•
Q
+
,
R
+
: the positive rationals/reals
5
Set Notation
• 
S

=
cardinality of
(number of elements in)
S
◦ {
a, b, c
}
= 3
•
Subset
:
A
⊂
B
if every element of
A
is an element
of
B
◦
Note:
Lots of people (including me, but not the
authors of the text) usually write
A
⊂
B
only if
A
is a
strict
or
proper
subset of
B
(i.e.,
A
6
=
B
).
I write
A
⊆
B
if
A
=
B
is possible.
•
Power set:
P
(
S
) is the set of all subsets of
S
(some
times denoted 2
S
).
◦
E.g.,
P
(
{
1
,
2
,
3
}
) =
{∅
,
{
1
}
,
{
2
}
,
{
3
}
,
{
1
,
2
}
,
{
1
,
3
}
,
{
2
,
3
}
,
{
1
,
2
,
3
}}
◦ P
(
S
)

= 2

S

6
Set Operations
•
Union:
S
∪
T
is the set of all elements in
S
or
T
◦
S
∪
T
=
{
x

x
∈
S
or
x
∈
T
}
◦ {
1
,
2
,
3
} ∪ {
3
,
4
,
5
}
=
{
1
,
2
,
3
,
4
,
5
}
•
Intersection:
S
∩
T
is the set of all elements in both
S
and
T
◦
S
∩
T
=
{
x

x
∈
S, x
∈
T
}
◦ {
1
,
2
,
3
} ∩ {
3
,
4
,
5
}
=
{
3
}
•
Set Diference:
S

T
is the set of all elements in
S
not in
T
◦
S

T
=
{
x

x
∈
S, x /
∈
T
}
◦ {
3
,
4
,
5
}  {
1
,
2
,
3
}
=
{
4
,
5
}
•
Complementation:
S
is the set of elements not in
S
◦
What is
{
1
,
2
,
3
}
?
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 Spring '07
 SELMAN
 Computer Science

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