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Discrete Mathematics II

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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

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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

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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. N m is the set of the first m positive numbers i.e. {1,2,3,4,......,m}. N 0 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,......}

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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={a 1,...,a n } a1a1 n choices n-1 choices 1 choice a2a2 anan 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

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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

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

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

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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

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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 nn-1n-k-2n-k-1 no. of choices 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) = C n k = n!/(n-k)!k!

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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

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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

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Groups A grouporis a set G with binary operationwhich satisfies the following properties 1.is a closed operation i.e. ifand 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 S n 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) ).

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Two show that an algebraic system is a group we must show that it satisfies all the axioms of a group. Question: Letbe a Boolean algebra so that A is a set of propositional elements,is like or,is like and andis like not. Show thatis 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 ?

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Group of Symmetries of a Triangle Consider the triangle X YZ O l 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

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Letdenote transformation y followed by transformation x for x and y in {1,p,q,a,b,c} So for example l m Yn O X Z l mn O l mn O X YZ XZ Y a p Notice the table is not symmetric

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Abelian Groups Ifis a group andis also commutative thenis referred to as an Abelian group (the name is taken from the 19th century mathematician N.H. Abel) is commutativemeans that Examples:andare abelian groups. Why isnot a group at all? If thenis an abelian group and is usually referred to as the group of integers modulo n

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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. ifand 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?

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Monoid is a monoid if the following conditions are satisfied: 1.is a closed operation i.e. ifand then 2.is associative 3. There is an identity element Examples: Let A be a finite set of heights. Let bea binary operation such that is equal tothe 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

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Properties of Algebraic Structures properties 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: ??

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Properties of Groups Theorem (The cancellation laws): Letbe a group then (i) (ii) Proof: (i) Suppose thatthenby axiom 3 a has an identityand we have that (ii) is proved similarly Theorem (The division laws): Letbe a group then (i) (ii) Proof ??

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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 ??

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For a an arbitrary element of a group we can define functionsand 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.

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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. Letthen which is the right coset with respect to

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Isomorphism Two groups are isomorphic if there is a bijection of one onto the other which preserves the group operations i.e. ifandare 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.

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

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Subgroups is a subgroup of the groupif andis also a group Examples:is a subgroup of Test for a subgroup Let H be a subset of G. Thenis a subgroup ofiff the following conditions all hold: (1) (2) H is closed under multiplication (3) For every group,andare subgroups is called the trivial subgroup of a proper subgroup ofis a subgroup different from G A non-trivial proper subgroup is a subgroup equal neither to or to

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

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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; Letbe an algebraic system with two binary operations and. Then the operation is said to distribute over the operationif and Example:distributes over + distributes over

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Ring An algebraic systemis called a ring if the following conditions are satisfied: (1)is an Abelian group (2)is a semigroup (3) The operationis 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 whichis commutative A ring with unity contains an element 1 such that Example: the ring of 2x2 matrices under matrix addition and multiplication is a ring with unity. The element 1=I= (0 is the identity of )

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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 operationis distributive over the operation Example:The set of real numbers with respect to + and is a field. is not a field. Why?

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

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

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(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:

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