1. Discrete Mathematics II

2. Contents 1 Introduction
2 Combinatorics, permutations and combinations.
3 Algebraic Structures and matrices: Homomorphism, commutative diagrams, isomorphism, semigroup, monoid, rings and fields
4 Vector Spaces
5 Lattice and Boolean algebras

3. Introduction Computer programs frequently handle real world data.
This data might be financial e.g. processing the accounts of a company.
It may be engineering data e.g. from sensors or actuators in a robotic system.
It may be scientific data e.g. weather data or geological data concerning rock strata.
In all these cases data typically consists of a set of discrete elements.
Furthermore there may exist orderings or relationships among elements or objects.
It may be meaningful to combine objects in some way using operators.
We hope to clarify our concepts of orderings and relationships among elements or objects
We look at the idea of formal structures such as groups , rings and and formal systems such as lattices and Boolean algebras

4. Number Systems
The set of natural numbers is the infinite set of the positive integers. It is denoted N and can have different representations:
{1,2,3,4, }
{1,10,11,100,101,.....}
are alternative representations of the same set expressed in different bases. Nm is the set of the first m positive numbers i.e. {1,2,3,4, ,m}. N0 is the set of natural numbers including 0 i.e. {0,1,2,3,5,....}
Q denotes the set of rational numbers i.e. signed integers and fractions
{0,1,-2,2,-3,3,-3,....,1/2,-1/2,3/2,-3/2,5/2,
-5/2,....,1/3,-1/3,2/3,-2/3, }
R is the set of real numbers i.e. the coordinates of all the points on a line.
Z is the set of all integers, both positive and negative {.....,-3,-2,-1,0,1,2,3,......}

5. 2 Combinatorics: Permutations
A permutation of the elements of a set A is a bijection from A onto itself.
If A is finite we can calculate the number of different permutations. Suppose A={a1,...,an} n
choices
n-1
choices 1
choice a1 a2 an
total number of ways of filling the n boxes
n x (n-1)x(n-2)x(n-3) x1=n!
nPn=n!
eg a possible permutation of {1,2,3,4,5,6} is

6. Composition of Permutations
If :A A and :A A are permutations of A then the composition or product .of  and satisfies for all x in A
.x)= (x))
Notice that since both and are bijections from A into A so is . In other words . is a permutation of A.
Example: Let A={1,2,3,4,5,6} then two possible permutations are
For . we have that

7. Cyclic Permutations A cyclic permutation on a set A of n elements has
the form where :
For shorthand we often write
 is said to be a k cycle
Example
or (6 1 4) is a cyclic
permutation
Two cyclic permutations
and
are said to be disjoint if
e.g. (4 5 2) and (3 1 6) are disjoint

8. Notice that
Other examples are or
Can you spot a product of disjoint cyclic
permutations equivalent to the following
permutation ?

9. Theorem: Every permutation of a finite set A can be expressed as a combination of disjoint cycles.
Structure underlying permutations
Note that the following hold:
(1) The product of two permutations is a uniquely determined permutation of the same set.
(2) The composition of permutations is associative.
(3) The permutation
is called the identity permutation and has the
property that
(4) For every permutation
there is an inverse
such that

10. Combinations
When we think about combinations we do not allow repeats and unlike permutations we do not consider order.
Combinations look at the number of different ways of picking a subset of k elements from a set of n elements.
Think of the number of ways of picking a list of k distinct elements of n
no. of choices n
n-1
n-k-2
n-k-1
places
= n(n-1)(n-2) (n-k-1) = n!/(n-k)!
For each possible list there are k! permutations
so since we are not interested in order we
should divide the above by k!.
C(n,k) = Cnk = n!/(n-k)!k!

11. Example: Choosing 2 elements from {a,b,c,d}
{a,b},{a,c},{a,d},
{b,c},{b d},{c,d}
C(4,2)= 4!/(2! 2!) =6
Combinations with Repetitions
We could also consider combinations with repetitions. With repetitions the number of distinct combinations of k elements chosen from n is:
C(n+k-1,k)= (n+k-1)!/k!(n-1)!
Number of different throws of 2 identical dice
(1 1)(2 2)(3 3)(4 4)(5 5)(6 6)
(1 2)(1 3)(1 4)(1 5)(1 6)
(2 3)(2 4)(2 5)(2 6)
(3 4)(3 5)(3 6)(4 5)(4 6)(5 6)
C(7,2)=21

12. Algebraic Structures
When we consider the behaviour of permutations under the composition operation we noticed certain underlying structures.
Permutations are closed under this operation, they exhibit associativity, an identity element exists and an inverse exists for each permutation
These properties define a general type of algebraic structure called a group.
In this section we shall look at groups in more detail as well as other similar algebraic structures such as semigroups and monoids.
Later we will progress to consider more complex algebraic structures such as rings, integral domains and fields.
We will see that many real life situations are examples of these algebraic structures

13. Groups A group or is a set G with binary operation
which satisfies the following properties 1.
is a closed operation i.e. if
and
then 2.
this is the
associative law
3. G has an element e, called the identity, such that 4.
there corresponds an element
such that
Examples: (1) The set of all permutations of a set A
onto itself is group (called the symmetric group Sn
for n elements).
(2) The set consisting of all (nxn) matrices that have
inverses is a group under ordinary matrix
multiplication( it is called GL(n) ).

14. Two show that an algebraic system is a group we
must show that it satisfies all the axioms of a group.
Question: Let
be a Boolean algebra
so that A is a set of propositional elements, is
like ‘or’,
is like ‘and’ and
is like ‘not’. Show
that
is an abelian group where
Answer:
(1) Associative since
prove this ?
(2) Has an identity element 0 (false) since
(3) Each element is its own inverse
(4) The operation commutes
prove this ?

15. Group of Symmetries of a Triangle
Consider the triangle X O Y Z n m
We can perform the following transformations
on the triangle
1=identity mapping from the plane to itself
p=rotation anticlockwise about O through 120
degrees
q=rotation clockwise about O through 120 degrees
a=reflection in l
b=reflection in m
c=reflection in n

16. Let
denote transformation y followed by
transformation x for x and y in {1,p,q,a,b,c}
So for example l l l Y X X a p O O O X Z m n
m Y Z n m Z Y n
Notice the table is not symmetric

17. Abelian Groups If is a group and is also commutative then
is referred to as an Abelian group
(the name is taken from the 19’th century
mathematician N.H. Abel)
is commutative
means that
Examples:
and
are
abelian groups.
Why is
not a group at all? If
then
is an abelian group and is usually
referred to as the group of integers modulo n

18. Semigroup An Abelian group is a strengthening of the notion of
group (i.e. requires more axioms to be satisfied)
We might also look at those algebraic structures
corresponding to a weakening of the group axioms
is a semigroup if the following conditions
are satisfied: 1.
is a closed operation i.e. if
and
then 2.
is associative
Example: The set of positive even integers
{2,4,6,.....} under the operation of ordinary addition
since
The sum or two even numbers is an even number
+ is associative
The reals or integers are not semigroups under -
why?

19. Monoid is a monoid if the following conditions are satisfied: 1.
is a closed operation i.e. if
and
then 2.
is associative
3. There is an identity element
Examples: Let A be a finite set of heights. Let be
a binary operation such that
is equal to
the taller of a and b. Then
is a monoid where the identity is the shortest
person in A
is a monoid:
is associative,
true is the identity, but false has no inverse
is a monoid:
is associative
false is the identity, but true has no inverse

20. Properties of Algebraic Structures
Theorem: (unique identity) Suppose that
is a monoid then the identity element is unique
Proof: Suppose there exist two identity elements
e and f. [We shall prove that e=f]
Theorem: (unique inverse) Suppose that
is a monoid and the element x in A has an inverse.
Then this inverse is unique.
Proof: ??

21. Properties of Groups Theorem (The cancellation laws): Let be
a group then
(i)
(ii)
Proof: (i) Suppose that
then
by axiom 3
a has an identity
and we have that
(ii) is proved similarly
Theorem (The division laws): Let be
a group then
(i)
(ii)
Proof ??

22. Theorem (double inverse) :If x is an element of
the group then
Proof:
Theorem (reversal rule)
If x and y are elements of the group then
Proof ??

23. we
For a an arbitrary element of a group
can define functions
and
such that
Theorem:
and
are permutations of G
Proof: Consider
[prove 1-1] suppose for x,y in G
[Prove onto] For any y in G
Corollary: In every row or column of the
multiplication table of G each element of G appears
exactly once.

24. Cosets . Consider a set A with a subset H. Let
Then the left coset of H with respect to a is
the set of elements:
This is denoted by
Similarly the right coset of H with respect to a is
and is denoted by
Example: Let A be the set of rotations
and
. Let
then
which is the right coset with respect to

25. Isomorphism
Two groups are isomorphic if there is a bijection of one onto the other which preserves the group operations i.e. if
and
are groups then a bijection
is an isomorphism provided
Example: Consider the group of matrices
of the form where under matrix
multiplication. This is isomorphic to the group
The mapping is
An isomorphism from a group onto itself is
called an automorphism.

26. Homomorphisms
The idea of isomorphic algebraic structures can be readily generalised by dropping the requirement that the functional mapping be a bijection.
Let
and
be two algebraic systems
then a homomorphism from to
is a functional mapping
such that
Example: consider the two structures
then f such that
is a homomorphism between
and

27. Subgroups is a subgroup of the group if and is also a group Examples:
Test for a subgroup
Let H be a subset of G. Then
is a subgroup of
iff the following conditions all hold:
(1)
(2) H is closed under multiplication
(3)
For every group ,
and
are
subgroups
is called the trivial subgroup of
a proper subgroup of
is a subgroup
different from G
A non-trivial proper subgroup is a subgroup
equal neither to or to

28. Normal Subgroups Let be a subgroup of . Then
is a normal subgroup if, for any
, the left
coset
is equal to the right coset
is a normal subgroup where
e.g.
Theorem: In an Abelian group, every subgroup
is a normal subgroup

29. Algebraic Structures with two Operations
So far we have studied algebraic systems with one binary operation. We now consider systems with two binary operations.
In such a system a natural way in which two operations can be related is through the property of distributivity;
Let
be an algebraic system with two
binary operations
and
. Then the operation
is said to distribute over the operation if
and
Example:
distributes over +
distributes over
distributes over

30. Ring An algebraic system is called a ring if
the following conditions are satisfied:
(1)
is an Abelian group
(2)
is a semigroup
(3) The operation
is distributive over the
operation
Example:
is a ring since
is an Abelian group
is a semigroup
distributes over +
A commutative ring is a ring in which is
commutative
A ring with unity contains an element 1 such
that
(0 is the identity of )
Example: the ring of 2x2 matrices under matrix
addition and multiplication is a ring with unity.
The element 1=I=

31. Integral Domains and Fields
is an integral domain if it is a commutative
ring with unity that also satisfies the following
property;
is also an integral domain
is a field if:
(1)
is an Abelian group
(2)
is an Abelian group
(3) The operation
is distributive over the
operation
Example:The set of real numbers with respect to
+ and is a field.
is not a field. Why?

32. A Field is an Integral Domain
Let
be a field then certainly
is a commutative ring with unity. Hence, it only
remains to prove that
Now suppose
then if x=0 the above
holds. Consider the case then where
Since
is an Abelian group then it
must contain an inverse to x,
, for which the
following holds
Now
Therefore y=0 as required

33. Properties Theorem: if is a ring. Then Proof: as for previous argument
Let -x denote the inverse of x under
Theorem: if
is a ring then the following
hold
(i)
(ii)
Proof: (i)

34. (ii)
for both (i) and (ii) the symmetric cases are
proved similarly
Theorem: suppose that elements a,b and c of
an integer domain satisfy and
then b=c.
Proof: