bered list
1.
2.
3.
4.
Apples
mangoes
lemons
bananas
Numbered list
A.
B.
C.
D.
Apples
mangoes
lemons
bananas
a. Apples
b. mangoes
c. lemons
d. bananas
I. Apples
II. mangoes
III. lemons
IV. bananas
i. Apples
ii. mangoes
iii. lemons
iv. bananas
Build your toolbox of abstract
structures and concepts.
Know the capacities and
limits of each tool.
groups
3 x 7
3 (3 x ) 3 7
( 3 3) x 4
0 x 4
x 4
Inverse, Closure
Associativity
Inverse
Identity
What makes this calculation possible are
abstract properti
Permutation Groups
Definition
A permutation of a set A is a function
from A to A that is both one to one and
onto.
Array notation
Let A = cfw_1, 2, 3, 4
Here are two permutations of A:
1 2 3 4
2 3 1 4
1 2 3 4
2 1 4 3
(2) 3
(4) 4
(2) (3) 4
Symmetric groups, Sn
Let A = cfw_1, 2, n. The symmetric
group on n letters, denoted Sn, is the
group of all permutations of A under
composition.
Sn is a group for the same reasons that
S3 is group.
|Sn| = n!
Symmetries of a square, D4
1
R0
1
1
R90
2
1
R
Theorem:
*
n
Z
with multiplication modulo n is a group
Proof:
associativity.
identity.
closure: Let * denote multiplication modulo n. Suppose (a*b, n)
> 1. Then there is a prime p such that p | n and p |
a*b.
p | a*b p | ab-kn, for some k (note: ab is NOT
3 x 7
3 (3 x ) 3 7
( 3 3) x 4
0 x 4
x 4
Inverse, Closure
Associativity
Inverse
Identity
What makes this calculation possible are
abstract properties of integers and
addition.
Closure: the sum of two integers is an integer
Associativity: (x + y) + z = x +
Isomorphism
Mapping between objects, which
shows that they are
structurally identical.
Any property which is preserved
by an isomorphism and which is
true for one of the objects, is
also true of the other.
Isomorphism
Example.
cfw_1,2,3,, or cfw_I, II, II
Algebraic Structures:
Group Theory II
Group
A group G is a pair (S,), where S is a
set and is a binary operation on S
such that:
1. is associative
2. (Identity) There exists an
element e S such that:
e a = a e = a,
for all a S
3. (Inverses) For every a S
Main Topics
The meaning of Object Orientation
Encapsulation
object identity
Information-hiding
Polymorphism
Generosity
Object-oriented programming
We present a rich style in program structure based on a
collection of stateful entities (abstract data type
Algebraic Structures:
Groups, Rings, and Fields
The RSA Cryptosystem
Rivest, Shamir, and Adelman (1978)
RSA is one of the most used cryptographic
protocols on the net. Your browser uses it
to establish a secure session with a site.
Zn = cfw_0, 1, 2, , n-1
Algebraic Structures:
Groups, Rings, and Fields
The RSA Cryptosystem
Rivest, Shamir, and Adelman (1978)
RSA is one of the most used cryptographic
protocols on the net. Your browser uses it
to establish a secure session with a site.
Zn = cfw_0, 1, 2, , n-1
Set Theory (Continued)
1
Disjointedness
Two sets A, B are called
disjoint (i.e., unjoined)
iff their intersection is
empty. (AB=)
Example: the set of even
integers is disjoint with
the set of odd integers.
2
Help, Ive
been
disjointed!
Inclusion-Exclusion
Mathematical Induction
(Continued)
1
Mathematical Induction:
Example
Show that any postage of 8 can be
obtained using 3 and 5 stamps.
First check for a few particular values:
8
=
3 + 5
9
=
3 + 3 + 3
10
=
5 + 5
11
=
5 + 3 + 3
12
=
3 + 3 + 3 + 3
How to g
Relations
And
Functions
A relation is a set of ordered pairs.
The domain is the set of all x values in the relation
domain = cfw_-1,0,2,4,9
These are the x values written in a set from smallest to largest
cfw_(2,3), (-1,5), (4,-2), (9,9), (0,-6)
These are
Functions
A relation is a set of ordered pairs.
The domain is the set of all x values in the relation
domain = cfw_-1,0,2,4,9
These are the x values written in a set from smallest to largest
cfw_(2,3), (-1,5), (4,-2), (9,9), (0,-6)
These are the y values
Finite Fields
The next morning at daybreak, Star flew indoors,
seemingly keen for a lesson. I said, "Tap eight." She did
a brilliant exhibition, first tapping it in 4, 4, then giving me
a hasty glance and doing it in 2, 2, 2, 2, before coming
for her nut.
Set Theory
1
Sets and subsets
Definitions
Element and set , Ex 3.1
Finite set and infinite set, cardinality A
, Ex 3.2
CD a subset, CD a proper subset
C=D, two sets are equal
Neither order nor repetition is relevant
for a general set
null set, cfw_
Discrete
Mathematics
Equivalence Relations
Introduction
Certain combinations of relation properties are very
useful
We wont have a chance to see many applications in
this course
In this set we will study equivalence relations
A relation that is reflex