Discrete Mathematics II

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

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

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,......}

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

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

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

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

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

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!

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

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

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) ).

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 ?

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

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

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

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?

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

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: ??

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 ??

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 ??

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.

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

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.

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

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

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

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

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=

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?

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

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)

(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: