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
differ 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
•
Finding 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
First, 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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Important 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
negationslash
= 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
negationslash
=
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 Difference:
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
}
?
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '05
 KLEINBERG
 Computer Science, Equivalence relation, Transitive relation

Click to edit the document details