Sets and subsets
A set is a clearly defined collection of objects known as its elements However, caution is necessary when using the term "set" due to Bertrand Russell's paradox, which illustrates that not all collections can be considered sets Russell examined the collection of all sets that are not elements of themselves, leading to a contradiction regarding whether this collection is an element of itself This raises concerns about the validity of a set being an element of itself, reinforcing the importance of the term "well-defined." Collections that do not meet this criterion are referred to as proper classes.
In set theory, sets are represented by capital letters, while their elements are indicated by lowercase letters The notation a∈A signifies that element a belongs to set A, whereas a /∈A indicates that a is not an element of A Sets can be defined by explicitly listing their elements within braces, such as {a, b, c, d}, or through a formal description outlining the characteristics of the elements.
A= {a|ahas propertyP}, i.e.,Ais the set of all objects with the propertyP IfAis a finite set, the number of its elements is written
Subsets LetAandBbe sets If every element ofAis an element ofB, we write
A⊆B and say thatAis asubsetofB, or thatAis containedinB IfA⊆BandB⊆A, so thatAandBhave exactly the same elements, thenAandBare said to beequal,
The negation of this isA =B The notationA ⊂B is used ifA ⊆ BandA =B; thenAis apropersubset ofB.
An empty set, denoted as E, is defined as a set that contains no elements It is considered a subset of any set A, as there cannot be an element in E that is not also in A Consequently, there is only one unique empty set; if there were two empty sets, E and E', they would still be equal since both are subsets of each other This unique empty set is universally recognized in set theory.
Some further standard sets with a reserved notation are:
N, Z, Q, R, C, which are respectively the sets of natural numbers 0,1,2, ,integers, rational num- bers, real numbers and complex numbers.
Set operations are fundamental concepts in mathematics, particularly union, intersection, and complement For sets A and B, the union (A ∪ B) includes all elements that belong to either A or B, while the intersection (A ∩ B) contains only those elements that are common to both sets.
It should be clear how to define the union and intersection of an arbitrary collection of sets{A λ |λ∈ }; these are written λ ∈
Frequently one has to deal only with subsets of some fixed setU, called theuniversal set IfA⊆U, then thecomplementofAinU is
Properties of set operations We list for future reference the fundamental properties of union, intersection and complement.
(1.1.1) LetA, B, Cbe sets Then the following statements are valid:
(vi) A− λ ∈ B λ λ ∈ (A−B λ )andA− λ ∈ B λ λ ∈ (A−B λ ) (De Morgan’s Laws) 1
The easy proofs of these results are left to the reader as an exercise: hopefully most of these properties will be familiar.
In the context of set theory, let A1, A2, , An represent distinct sets An n-tuple is defined as a sequence of elements a1, a2, , an where each element ai belongs to the corresponding set Ai Typically, this n-tuple is expressed in the format (a1, a2, , an), and the collection of all possible n-tuples is referred to as the set of n-tuples.
This is theset product(orcartesian product) ofA 1 , A 2 , , A n For exampleR×R is the set of coordinates of points in the plane.
The following result is a basic counting tool.
(1.1.2) IfA 1 , A 2 , , A n are finite sets, then|A 1 ìA 2 ìã ã ãìA n | = |A 1 |ã|A 2 | .|A n |.
Proof In forming an n-tuple (a 1 , a 2 , , a n ) we have |A 1| choices for a 1, |A 2| choices fora 2 , ,|A n |choices fora n Each choice ofa i ’s yields a differentn-tuple.
Therefore the total number ofn-tuples is|A 1 | ã |A 2 | .|A n |.
The power set Thepower setof a setAis the set of all subsets ofA, including the empty set andAitself; it is denoted by
The power set of a finite set is always a larger set, as the next result shows.
Proof LetA= {a 1 , a 2 , , a n }with distincta i ’s Also putI = {0,1} Each subset
B ofAis to correspond to ann-tuple(i 1 , i 2 , , i n )withi j ∈I Here the rule for forming then-tuple corresponding toBis this:i j =1 ifa j ∈Bandi j =0 ifa j ∈/B.
Conversely every n-tuple(i 1 , i 2 , , i n )with i j ∈ I determines a subset B of A, defined byB = {a j | 1≤j ≤n, i j =1} It follows that the number of subsets of
Aequals the number of elements inIìIì ã ã ã ìI, (withnfactors) By (1.1.2) we obtain|P (A)| =2 n =2 | A |
The power setP (A), together with the operations ∪ and∩, constitute what is known as aBoolean 2 algebra; such algebras have become very important in logic and computer science.
1 Prove as many parts of (1.1.1) as possible.
2 Let A, B, Cbe sets such thatA∩B = A∩CandA∪B = A∪C Prove that
3 IfA, B, Care sets, establish the following:
4 Thedisjoint unionA⊕Bof setsAandBis defined by the ruleA⊕B=A∪B−A∩B, so its elements are those that belong to exactly one ofAandB Prove the following statements:
Relations, equivalence relations and partial orders
In mathematics, understanding the relationships between elements of a set is often crucial, rather than just focusing on the individual elements themselves This necessity gives rise to the concept of a relation, which describes how these elements are interconnected.
LetAandBbe sets Then arelation R betweenAandBis a subset of the set productA×B The definition will be clarified if we use a more suggestive notation: if(a, b)∈R, thenais said to berelatedtobbyRand we write a R b.
The most important case is of a relationRbetweenAand itself; this is calleda relation on the setA.
Examples of relations (i) LetAbe a set and defineR = {(a, a)| a ∈ A} Thus a 1 R a 2 means thata 1 =a 2 andRis the relation of equality onA.
(ii) LetP be the set of points andLthe set of lines in the plane A relationRfrom
P toLis defined by: p R if the pointplies on the line SoR is the relation of incidence.
(iii) A relationRon the set of integersZis defined by:a R bifa−bis even. The next result confirms what one might suspect, that a finite set has many relations.
(1.2.1) IfAis a finite set, the number of relations onAequals2 | A | 2
For this is the number of subsets ofA×Aby (1.1.2) and (1.1.3).
A relation on a set is a broad concept, but the most interesting relations are those that possess specific properties The most common types of relations, denoted as R on a set A, are highlighted in this discussion.
Equivalence relations are defined by their reflexive, symmetric, and transitive properties, making them essential in various mathematical contexts Similarly, partial orders are significant relations characterized by being reflexive, antisymmetric, and transitive.
Examples (a) Equality on a set is both an equivalence relation and a partial order.
(b) A relationRonZis defined by: a R bif and only ifa−bis even This is an equivalence relation.
(c) IfAis any set, the relation of containment⊆is a partial order on the power set
(d) A relationRonNis defined bya R bifadividesb HereRis a partial order onN.
Equivalence relations on a set lead to the formation of distinct, non-overlapping subsets This analysis reveals that an equivalence relation effectively partitions the set into non-empty subsets, highlighting the structured nature of these relationships.
LetEbe an equivalence relation on a setA First of all we define theE-equivalence classof an elementaofAto be the subset
By the reflexive lawa∈ [a] E , so
[a] E andAis the union of all the equivalence classes.
Next suppose that the equivalence classes[a] E and[b] E both contain an integerx.
Assume thaty ∈ [a] E ; theny E a,a E x andx E b, by the symmetric law Hence y E b by two applications of the transitive law Therefore y ∈ [b] E and we have proved that[a] E ⊆ [b] E By the same reasoning[b] E ⊆ [a] E , so that[a] E = [b] E
It follows that distinct equivalence classes are disjoint, i.e., they have no elements in common.
Set A is formed by the union of E-equivalence classes, which are distinct and disjoint A partition of A refers to a decomposition of A into non-empty, disjoint subsets, and thus, E establishes a partition of A.
To construct an equivalence relation on a set A based on a given partition into non-empty disjoint subsets A λ, we define the relation a E b to indicate that elements a and b belong to the same subset A λ This definition ensures that E is an equivalence relation, and the equivalence classes correspond precisely to the original subsets A λ of the partition.
We summarize these conclusions in:
(1.2.2) (i)IfEis an equivalence relation on a setA, theE-equivalence classes form a partition ofA.
(ii)Conversely, each partition ofAdetermines an equivalence relation on A for which the equivalence classes are the subsets in the partition.
Thus the concepts of equivalence relation and partition are in essence the same.
Example (1.2.1) In the equivalence relation (b) above there are two equivalence classes, the sets of even and odd integers; of course these form a partition ofZ.
Partial orders Suppose thatRis a partial order on a setA, i.e., Ris a reflexive, antisymmetric, transitive relation onA Instead of writinga R bit is customary to employ a more suggestive symbol and write ab.
The pair(A,) then constitutes apartially ordered set(orposet).
The effect of a partial order is to impose a hierarchy on the setA This can be visualized by drawing a picture of the poset called aHasse 3 diagram It consists of
Helmut Hasse (1898-1979) contributed significantly to the understanding of relationships between elements in a set, represented through vertices and edges in a plane In this graphical representation, vertices symbolize elements of a set A, while upward sloping edges indicate a specific relationship, such as "a is related to b." Conversely, elements not connected by such edges do not satisfy this relationship To enhance clarity, it is essential to omit unnecessary edges in the diagram.
A very familiar poset is the power set of a set A with the partial order ⊆, i.e.
To draw the Hasse diagram of the poset (P(A), ⊆) where A = {1, 2, 3}, we first note that this poset consists of 2^3 = 8 vertices These vertices can be represented visually as the corners of a cube positioned on one of its vertices in a two-dimensional plane.
Partially ordered sets are important in algebra since they can provide a useful representation of substructures of algebraic structures such as subsets, subgroups, subrings etc
A linear order on a set A is defined as a partial order where, for any elements a and b in A, either a is related to b or b is related to a This structure, denoted as (A, ≤), is referred to as a linearly ordered set or chain, which can be visually represented by a Hasse diagram that shows a single upward-sloping sequence of edges Common examples of chains include the sets of integers (Z, ≤) and real numbers (R, ≤) under the standard "less than or equal to" relation Furthermore, a linear order on A is classified as a well order if every non-empty subset X of A has a least element a, meaning that a is less than or equal to all elements x in X An example of a well order is the relation ≤ on the set of positive integers, which is supported by the Well-Ordering Law, an axiom explored in Section 2.1.
In a poset (A, ≤), for any elements a and b in A, the least upper bound (lub) is defined as an element u in A such that a ≤ u and b ≤ u, and for any x in A satisfying a ≤ x and b ≤ x, it follows that u ≤ x Conversely, the greatest lower bound (glb) of a and b is an element g in A such that g ≤ a and g ≤ b, and for any x in A where x ≤ a and x ≤ b, it must be the case that x ≤ g The Hasse diagram of (A, ≤) visually represents these relationships, including the lub and glb, often depicted as a lozenge-shaped figure connecting a, b, and g.
A poset in which each pair of elements has an lub and a glb is called alattice For example,(P (S),⊆)is a lattice since the lub and glb ofAandBareA∪BandA∩B respectively.
In the study of relations, two relations can be combined through union or intersection, but a more effective method is composition When considering relations R and S, where R is a relation between sets A and B, and S is a relation between sets B and C, the composite relation is formed by linking these sets together.
SR is the relation betweenAandC defined bya SR cif there existsb∈B such that a R bandb S c.
For example, assume that A = Z, B = {a, b, c}, C = {α, β, γ} Define re- lations R = {(1, a), (2, b), (4, c)}, S = {(a, α), (b, γ ), (c, β)} Then S R {(1, α), (2, γ ), (4, β)}.
In particular one can form the composite of any two relationsRandSon a setA.
Notice that the condition for a relationRto be transitive can now be expressed in the formRR⊆R.
A result of fundamental importance is the associative law for composition of rela- tions.
(1.2.3) LetR, S, T be relations betweenAandB,BandC, andCandDrespectively.ThenT (SR)=(T S)R.
Proof Leta∈Aandd ∈D Thena (T (SR)) d means that there existsc ∈C such thata (SR) candc T d, i.e., there existsb ∈B such thata R b,b S cand c T d Thereforeb (T S ) danda ((T S)R ) d ThusT (SR)⊆(T S)R, and in the same way(T S)R⊆T (SR).
1 Determine whether or not each of the binary relationsRdefined on the setsAbelow is reflexive, symmetric, antisymmetric, transitive:
(d)A=Zanda R bmeans thatb=a+3cfor some integerc.
2 A relation∼onR−{0}is defined bya∼bifab >0 Show that∼is an equivalence relation and identify the equivalence classes.
3 LetA= {1,2,3, ,12}and letabmean thatadividesb Show that(A,)is a poset and draw its Hasse diagram.
4 Let(A,)be a poset and leta, b∈A Show thataandbhave at most one lub and at most one glb.
5 Given linearly ordered sets(A i , i ), i =1,2, , k, suggest a way to makeA 1×
A 2ì ã ã ã ìA k into a linearly ordered set.
6 How many equivalence relations are there on a setSwith 1,2,3 or 4 elements?
7 Suppose thatAis a set withnelements Show that there are 2 n 2 − n reflexive relations onAand 2 n(n + 1)/2 symmetric ones.
Functions
Functions are fundamental concepts in mathematics, often first introduced in calculus as real-valued functions of a real variable They serve as useful tools for describing intricate processes in both mathematics and information sciences.
LetAandBbe sets Afunctionormapping fromAtoB, in symbols α:A→B, is a rule which assigns to each elementaofAa unique elementα(a)ofB, called the image ofaunderα The setsAandBare thedomainandcodomainofαrespectively. Theimageof the functionαis
Im(α)= {α(a)|a∈A}, which is a subset ofB The set of all functions fromAtoBwill sometimes be written Fun(A, B).
Functions in calculus are defined as those with domains and codomains that are subsets of the real numbers (R) These functions can be effectively represented by graphing them in the conventional manner.
(b) Given a functionα:A→B, we may define
ThenR α is a relation betweenAandB Observe thatR α is a special kind of relation since eachainAis related to a unique element ofB, namelyα(a).
In a scenario where R is a relation between sets A and B, and each element a in A corresponds to a unique element b in B, we can define a function α R: A → B This function is expressed as α R(a) = b if and only if a R b Therefore, functions from A to B can be viewed as specific types of relations between these two sets.
This observation permits us to form the composite of two functionsα : A →B and β : B → C, by forming the composite of the corresponding relations: thus βα:A→Cis defined by βα(a)=β(α(a)).
(c)The characteristic function of a subset LetAbe a fixed set For each subset
XofAdefine a functionα X :A→ {0,1}by the rule α X (a) 1 ifa∈X
Then α X is called the characteristic function of the subsetX Conversely, every functionα : A → {0,1}is the characteristic function of some subset ofA– which subset?
(d) The identity function on a setA is the function id A : A → A defined by id A (a)=afor alla∈A.
Injectivity and surjectivity are two essential types of functions in mathematics A function α: A → B is considered injective (or one-to-one) if it ensures that distinct elements in set A map to distinct images in set B, meaning that if α(a) = α(a'), then a must equal a' On the other hand, a function is surjective (or onto) if every element in set B is represented as the image of at least one element from set A, which can be expressed as Im(α) = B When a function α: A → B is both injective and surjective, it is referred to as bijective (or a one-to-one correspondence).
In the study of functions, consider the example of the function α: R → R defined by α(x) = 2x, which is injective but not surjective Conversely, the function α: R → R defined by α(x) = x³ - 4x is surjective, as demonstrated by its graph, where any horizontal line intersects the curve at least once However, this function is not injective, since it yields the same output for different inputs, such as α(0) = 0 and α(2) = 0.
(d)α:R→Rwhereα(x)=x 2 is neither injective nor surjective.
Inverse functions are defined as mutually inverse when two functions, α: A → B and β: B → A, satisfy the conditions αβ = id B and βα = id A If β serves as another inverse for α, it must equal β, as demonstrated by the associative law, confirming that α can only have one unique inverse, denoted as α^(-1): B → A, if it exists.
It is important to be able to recognize functions which possess inverses.
(1.3.1) A functionα:A→Bhas an inverse if and only if it is bijective.
To prove that the function α is bijective, we start by assuming the existence of its inverse α − 1: A → B If α(a1) = α(a2), applying α − 1 to both sides yields a1 = a2, demonstrating that α is injective To establish that α is surjective, consider any element b in B; we can express b as idB(b) = α(α − 1(b)), which confirms that b is in the image of α Thus, we conclude that Im(α) = B, proving that α is surjective and, consequently, bijective.
Conversely, letα be bijective Ifb ∈ B, there is precisely one elementain A such thatα(a) = bsince α is bijective Defineβ : B → A byβ(b) = a Then αβ(b)=α(a)=bandαβ =id B Alsoβα(a)=β(b)=a; since everyainAarises in this way,βα=id A andβ=α − 1
The next result records some useful facts about inverses.
(1.3.2) (i)Ifα:A→Bis a bijective function, then so isα − 1 :B→A, and indeed (α − 1 ) − 1 =α.
(ii)Ifα:A→Bandβ:B→Care bijective functions, thenβ α:A→Cis bijective, and(βα) − 1 =α − 1 β − 1
Proof (i) The equationsαα − 1 =id B andα − 1 α=id A tell us thatαis the inverse ofα − 1
(ii) Check directly thatα − 1 β − 1 is the inverse ofβα, using the associative law twice: thus(βα)(α − 1 β − 1 )=((βα)α − 1 )β − 1 =(β(αα − 1 ))β − 1 (βid A )β − 1 =ββ − 1 =id C Similarly(α − 1 β − 1 )(βα)=id A
Application to automata As an illustration of how the language of sets and func- tions may be used to describe information systems we shall briefly consider automata.
An automaton is a theoretical model of a digital computer, featuring an input tape, an output tape, and a head that reads symbols from the input and prints symbols to the output The system operates in a specific state at any given moment, transitioning to a new state upon reading a symbol from the input tape and subsequently writing a symbol to the output tape.
To make this idea precise we define an automatonAto be a 5-tuple
An automaton is defined by the sets of input symbols (I), output symbols (O), and states (S), along with the output function (ν) and the next state function (σ) When the automaton reads an input symbol (i) from set I while in a current state (s) from set S, it outputs the corresponding symbol (ν(i, s)) on the output tape and transitions to a new state (σ(i, s)) This operation is governed by the interplay of the three sets and the two functions, determining the automaton's behavior.
1 Which of the following functions are injective, surjective, bijective?
2 Prove that the composite of injective functions is injective, and the composite of surjective functions is surjective.
3 Letα :A → Bbe a function between finite sets Show that if|A|> |B|, thenα cannot be injective, and if|A|0 Prove that there are integersu, v such that a=bu+vand− b 2 ≤v < b 2 [Hint: start with the Division Algorithm].
9 Prove that gcd{4n+5,3n+4} =1 for all integersn.
10 Prove that gcd{2n+6, n 2 +3n+2} =2 or 4 for any integern, and that both possibilities can occur.
11 Show that if 2 n +1 is prime, thennmust have the form 2 l (Such primes are calledFermat primes).
12 The only integernwhich is expressible asa 3 (3a+1)andb 2 (b+1) 3 witha, b relatively prime and positive is 2000.
Congruences
The notion of congruence was introduced by Gauss 2 in 1801, but it had long been implicit in ancient writings concerned with the computation of dates.
In number theory, two integers \(a\) and \(b\) are considered congruent modulo a positive integer \(m\), denoted as \(a \equiv b \, (\text{mod} \, m)\), if \(m\) divides the difference \(a - b\) This relationship defines an equivalence relation on the set of integers \(Z\), leading to the division of \(Z\) into distinct equivalence classes known as congruence classes modulo \(m\).
The unique congruence class to which an integerabelongs is written
By the Division Algorithm any integeracan be written in the forma =mq+r where q, r ∈ Z and 0 ≤ r < m Thusa ≡ r (modm)and[a] = [r] Therefore
[0],[1], ,[m−1]are all the congruence classes modulom Furthermore if[r] = [r ] where 0≤r, r < m, thenm||r−r , which can only mean thatr =r Thus we have proved:
(2.3.1) Letmbe any positive integer Then there are exactlymcongruence classes modulom, namely[0],[1], ,[m−1].
Congruence arithmetic We shall write
In the set of all congruence classes modulo \( m \), denoted as \( Z_m \), we can define operations of addition and multiplication for these congruence classes This allows us to perform arithmetic operations within \( Z_m \), thereby enriching the structure and functionality of congruence classes.
Thesumandproductof congruence classes modulomare defined by the rules
While the natural definitions of congruence classes are clear, it is important to exercise caution in their formulation Each class can be represented by any of its elements, so we must ensure that the defined sum and product are based solely on the congruence classes themselves, rather than on the specific representatives chosen.
To demonstrate the congruence relations, let \( a \in [a] \) and \( b \in [b] \) We aim to show that \( [a+b] = [a+b] \) and \( [ab] = [ab] \) By expressing \( a \) and \( b \) as \( a = a + mu \) and \( b = b + mv \) for some integers \( u, v \in \mathbb{Z} \), we derive that \( a + b = (a + b) + m(u + v) \) and \( ab = ab + m(av + bu + muv) \) Consequently, it follows that \( a + b \equiv a + b \mod m \) and \( ab \equiv ab \mod m \).
[a +b ] = [a+b], and[a b ] = [ab], which is what we needed to check.
Now that we know the sum and product of congruence classes to be well-defined, it is a simple task to establish the basic properties of these operations.
(2.3.2) Letmbe a positive integer and let[a],[b],[c]be congruence classes modulo m Then
Both Z and Z_m share common arithmetic properties, as outlined in section 2.1, which allows us to classify them as commutative rings with identity This classification will be further elaborated in Chapter 6.
Fermat's Theorem 3 is a significant mathematical principle that relates to binomial coefficients These coefficients, denoted as n choose r, represent the number of ways to select r objects from a set of n distinct objects, where n and r are integers satisfying the condition 0 ≤ r ≤ n The formula for calculating n choose r is widely recognized and frequently utilized in combinatorial mathematics.
The property we need here is:
Note that each prime appearing as a factor of the numerator or denominator ofmis smaller than p Writem = u v where uandv are relatively prime integers Then v p r
=pmv =puand by Euclid’s Lemmavdividesp Nowv=p, sov=1 and m=u∈Z Hencepdivides p r
We are now able to prove Fermat’s Theorem.
(2.3.4) Ifpis a prime andxis any integer, thenx p ≡x (mod p).
To prove that \( x^p \equiv x \mod p \), we can assume \( x \geq 0 \) without loss of generality, regardless of whether \( p \) is odd We will use mathematical induction on \( x \), starting with the base case where \( x = 0 \), which holds true Assuming the statement is valid for some integer \( x \), we can apply the Binomial Theorem to demonstrate that it also holds for \( x + 1 \).
(x+1) p p r = 0 p r x r ≡x p +1(mod p) since pdivides p r if 0 < r < p Therefore(x+1) p ≡ x+1(modp)and the induction is complete
Solving congruences involves finding unknown integers similar to solving equations for real numbers The basic form of a linear congruence with one unknown, x, is expressed as \( ax \equiv b \, (\text{mod} \, m) \), where a, b, and m are specified integers.
(2.3.5) Leta, b, mbe integers withm >0 Then there is a solutionxof the congruence ax ≡b (modm)if and only if gcd{a, m}dividesb.
Proof Setd =gcd{a, m} Ifxis a solution of congruenceax ≡ b (modm), then ax =b+mqfor someq∈Z; from this it follows at once thatdmust divideb.
Assuming that d is not equal to b, we can express d as a linear combination of integers u and v, such that d = au + mv By multiplying this equation by the integer b/d, we derive a new equation: b = a(ub/d) + m(vb/d) By letting x equal ub/d, we find that ax is congruent to b modulo m, indicating that x is a valid solution to the congruence.
The most important case occurs whenb=1.
Corollary (2.3.6) Leta, mbe integers withm > 0 Then the congruenceax ≡ 1
(mod m)has a solutionxif and only ifais relatively prime tom.
In congruence arithmetic, for any integer m greater than 0 and a congruence class [a] modulo m, there exists a congruence class [x] such that [a][x] equals [1] if and only if a is relatively prime to m This relationship allows us to identify which congruence classes modulo m possess inverses; specifically, these are the classes [x] where 0 < x < m and x is relatively prime to m The count of these invertible congruence classes modulo m is represented by the function φ(m).
Next let us consider systems of linear congruences.
(2.3.7)(The Chinese Remainder Theorem) Leta 1 , a 2 , , a k andm 1 , m 2 , , m k be integers withm i >0; assume thatgcd{m i , m j } =1ifi=j Then there is a common solutionxof the system of congruences
When k = 2, this striking result was discovered in the 1 st Century AD by the Chinese mathematician Sun Tse.
Proof of (2.3.7) Put m = m 1 m 2 m k and m i = m/m i Then m i and m i are relatively prime, so by (2.3.6) there exist an integer i such thatm i i ≡1(modm i ).
Now putx =a 1 m 1 1+ ã ã ã +a k m k k Then x≡a i m i i ≡a i (modm i ) sincem i ||m j ifi=j.
As an application of (2.3.7) a well-known formula for Euler’s function will be derived.
(2.3.8) (i)If m and n are relatively prime positive integers, thenφ (mn)=φ (m)φ (n).
(ii)Ifm=p 1 l 1 p 2 l 2 p k l k withl i >0and distinct primesp i , then φ (m) k i = 1 p i l i −p l i i − 1
The set of invertible congruence classes in \( Z_m \) is denoted as \( U_m \), with its size represented by \( |U_m| = \phi(m) \) A mapping \( \alpha: U_{mn} \rightarrow U_m \times U_n \) is defined by \( \alpha([a]_{mn}) = ([a]_m, [a]_n) \) It is important to note that this mapping is well-defined Furthermore, if \( \alpha([a]_{mn}) = \alpha([b]_{mn}) \), then it follows that the congruence classes are equal.
[a] m = [a ] m and[a] n = [a ] n , equations which imply thata−a is divisible bym andn, and hence bymn Thereforeαis an injective function.
The function α is surjective, meaning that for any given integers a in U_m and b in U_n, the Chinese Remainder Theorem guarantees the existence of an integer x that satisfies the congruences x ≡ a (mod m) and x ≡ b (mod n) Consequently, we have [x]_m = [a]_m and [x]_n = [b]_n, which leads to the conclusion that α([x]_{mn}) equals ([a]_m, [b]_n).
Thus we have proved thatαis a bijection, and consequently|U mn | = |U m | × |U n |, as required.
(ii) Suppose that pis a prime and n > 0 Then there arep n − 1 multiples ofp among the integers 0,1, , p n −1; thereforeφ (p n )=p n −p n − 1 Finally apply (i) to obtain the formula indicated.
We end the chapter with some examples which illustrate the utility of congruences.
Example (2.3.1) Show that an integer is divisible by 3 if and only if the sum of its digits is a multiple of 3.
Let n = a 0 a 1 a k be the decimal representation of an integern Thus n a k +a k − 1 10+a k − 2 10 2 + ã ã ã +a 0 10 k where 0≤a i s ≥ 0, such that πᵣ(i₁) = πₛ(i₁).
(π − 1 ) s to both sides of the equation and using associativity, we find thatπ r − s (i 1 ) i 1 Hence by the Well-Ordering Law there is a least positive integer m 1 such that π m 1 (i 1 )=i 1
We will demonstrate that the integers i₁, π(i₁), π²(i₁), , π^(m₁-1)(i₁) are all distinct If we assume that πᵣ(i₁) = πˢ(i₁) for m₁ > r > s ≥ 0, it follows that π^(r-s)(i₁) = i₁, which contradicts the condition 0 < r - s < m₁ Therefore, π permutes the m₁ distinct integers i₁, π(i₁), , π^(m₁-1)(i₁) in a cycle, confirming the existence of the m₁-cycle.
If π fixes all integers, it can be expressed as an m₁-cycle If not, there exists an integer i₂ that is not part of the first cycle, leading to the identification of a second disjoint cycle By repeatedly applying this process, π can be represented as a product of disjoint cycles To prove uniqueness, assume π has two representations as products of disjoint cycles Since disjoint cycles commute, we can align the cycles such that the first element remains the same, leading to identical representations for all elements Thus, the two expressions for π must be the same.
Corollary (3.1.4) Every element ofS n is expressible as a product of transpositions.
Proof Because of (3.1.3) it is sufficient to show that each cyclic permutation is a product of transpositions That this is true follows from the easily verified identity:
To express π as a product of transpositions first write it as a product of disjoint cycles, following the method of the proof of (3.1.3) Thusπ = (1 3 5 4)(2 6) Also(1 3 5 4) = (1 4)(1 5)(1 3), so that π =(1 4)(1 5)(1 3)(2 6).
On the other hand not every permutation inS n is expressible as a product ofdisjoint transpositions (Why not?)
In the context of permutations in S_n, a permutation π rearranges the natural order of integers 1, 2, , n into a new sequence π(1), π(2), , π(n) This rearrangement can lead to inversions, which are defined as instances where for indices i < j, the condition π(i) > π(j) holds true To effectively count the number of inversions created by the permutation π, a formal method is introduced.
Binary operations: semigroups, monoids and groups
In algebra, structures are primarily composed of a set along with specific rules for combining pairs of its elements We formalize these combination rules through the concept of a binary operation, which is defined as a function α: S × S → S, where S represents the set.
So for each ordered pair(a, b)witha, binSthe functionαproduces a unique element α((a, b))ofS It is better notation if we write a∗b instead ofα((a, b))and refer to the binary operation as∗.
Of course binary operations abound; one need think no further than addition or multiplication in sets such asZ,Q,R, or composition on the set of all functions on a given set.
The first algebraic structure of interest to us is asemigroup, which is a pair
(S,∗) consisting of a non-empty set S and a binary operation ∗ onS which satisfies the associative law:
If the semigroup has anidentity element,i.e., an elementeofS such that
(ii) a∗e=a=e∗afor alla∈S, then it is called amonoid.
Finally, a monoid is called agroupif each elementaofS has aninverse, i.e., an elementa − 1 ofSsuch that
Also a semigroup is said to becommutativeif
(iv) ab=bafor all elementsa, b.
A commutative group is called anabelian 2 group.
Thus semigroups, monoids and groups form successively narrower classes of al- gebraic structures These concepts will now be illustrated by some familiar examples.
Examples of semigroups, monoids and groups (i) The pairs (Z,+), (Q,+), (R,+) are groups where + is ordinary addition, 0 is an identity element and an inverse ofxis its negative−x.
In considering the sets (Q ∗, ã) and (R ∗, ã), where the dot represents standard multiplication, we find that both Q ∗ (the set of non-zero rational numbers) and R ∗ (the set of real numbers) form groups with an identity element of 1, and the inverse of any element x being 1/x Conversely, (Z ∗, ã) is classified as a monoid because certain integers, such as 2, lack an inverse within Z ∗, which is defined as Z minus the zero element.
(iii) (Z m ,+) is a group where m is a positive integer The usual addition of congruence classes is used here.
The group (Z ∗ m, ã) consists of invertible congruence classes [a] modulo m, where m is a positive integer and gcd{a, m} = 1, with multiplication of these classes defining the group operation The order of this group is given by |Z ∗ m| = φ(m), where φ represents Euler’s totient function Additionally, the set M n (R) encompasses all n×n matrices with real entries, and when the standard matrix addition is applied, it forms a group as well.
On the other hand,M n (R)with matrix multiplication is only a monoid To obtain a group we must form
GL n (R), the set of all invertible ornon-singularmatrices inM n (R), i.e., those with non-zero determinant This is called thegeneral linear group of degreenoverR.
An example of a semigroup that is not a monoid is the set of all even integers, denoted as (E, +), because it lacks an identity element.
(vii)The monoid of functions on a set LetAbe any non-empty set, and write Fun(A)for the set of all functionsαonA Then
(Fun(A),) is a monoid whereis functional composition; indeed this binary operation is asso- ciative by (1.2.3) The identity element is the identity function onA.
If we restrict attention to the bijective functions onA, i.e., to those which have inverses, we obtain thesymmetric grouponA
(Sym(A),), consisting of all the permutations ofA This is one of our prime examples of groups.
Monoids of words can be explored using an alphabet X, where X is a non-empty set A word in X is represented as an n-tuple of elements from X, conveniently written as x1 x2 xn, with n being greater than or equal to zero The special case where n equals zero corresponds to the empty word, denoted as ∅ The set of all words formed from the alphabet X is represented as W(X).
There is a natural binary operation on X, namely juxtaposition Thus ifw x 1 x n andz=y 1 y m are words inX, definewzto be the wordx 1 x n z 1 z m
In the context of free monoids, if the set W is empty (If W = ∅), the binary operation defined as wz = z = zw holds true by convention This operation is associative, and the empty set serves as the identity element Therefore, W(X), under this specified operation, forms a monoid known as the free monoid generated by the set X.
Monoids and state output automata share an intriguing relationship, as highlighted in section 1.3 Consider a state output automaton denoted as A = (I, S, ν), where I represents the input set, S is the state set, and ν is the next state function defined as ν: I × S → S.
ThenAdetermines a monoidM A in the following way.
Let \( \theta_i: S \rightarrow S \) be defined by the rule \( \theta_i(s) = \nu(i, s) \) for \( i \in I \) and \( s \in S \) The set \( M_A \) is formed by including the identity function along with all finite compositions of the \( \theta_i \) functions, making \( M_A \subseteq Fun(S) \) It is evident that \( M_A \) operates as a monoid under functional composition.
In fact one can go in the opposite direction as well Suppose we start with a monoid
(M,∗) and define an automaton A M = (M, M, ν) where the next state function ν : M ×M → M is defined by the ruleν(x 1 , x 2 ) = x 1∗x 2 Thus a connection between monoids and state output automata has been detected.
Symmetry groups play a crucial role in understanding the concept of symmetry, as they emerge in various contexts where symmetry is significant The size of a symmetry group serves as an indicator of the level of symmetry present in a given structure Given that symmetry is fundamentally a geometric concept, it is expected that geometry offers numerous compelling examples of these groups.
An isometry is a bijective function on 3-dimensional space or the plane that preserves distances between points Common examples of isometries include translations, rotations, and reflections In this context, let X represent a non-empty set of points in 3-space or the plane, referred to as a geometric configuration An isometry α that fixes the set X maintains the positions of all points within this configuration.
X= {α(x)|x∈X}, is called asymmetry ofX Note that a symmetry can move the individual points ofX.
It is easy to see that the symmetries ofXform a group with respect to functional composition; this is thesymmetry groupS(X)ofX ThusS(X)is a subset of Sym(X), usually a proper subset.
The symmetry group of the regular n -gon As an illustration let us analyze the symmetries of theregularn-gon: this is a polygon in the plane withnequal edges,
(n ≥3) It is convenient to label the vertices of then-gon 1,2, , n, so that each symmetry may be represented by a permutation of{1,2, , n}, i.e., by an element ofS n
Each symmetry arises from an axis of symmetry of the figure Of course, if we expect to obtain a group, we must include the identity symmetry, represented by
(1)(2) (n) There aren−1 anticlockwise rotations about the line perpendicular to the plane of the figure and through the centroid, through angles i 2π n
, for i 1,2, , n−1 For example, the rotation through 2π n is represented by then-cycle
(1 2 3 n); other rotations correspond to powers of thisn-cycle.
In a plane, reflections occur along axes of symmetry of an n-gon If n is odd, these axes connect a vertex to the midpoint of the opposite edge, exemplified by the sequence (1)(2n)(3n−1) Conversely, when n is even, there are two types of reflection axes: one joining opposite vertices and another connecting midpoints of opposite edges, resulting in a total of n reflections After considering all symmetry axes, we determine that the order of the symmetry group of the n-gon is 2n, known as the dihedral group of order 2n.
Notice that Dih(2n)is a proper subset ofS n if 2n < n!, i.e., ifn≥4 So not every permutation of the vertices arises from a symmetry whenn≥4.
Elementary consequences of the axioms We end this section by noting three ele- mentary consequences of the axioms.
The Generalized Associative Law states that for any elements \( x_1, x_2, \ldots, x_n \) in a semigroup \( (S, \ast) \), the element \( u \), formed by combining these elements in a specific order with any arrangement of parentheses, can be expressed as \( u = (x_1 \ast x_2) \ast x_3 \ast \ldots \ast x_n \) This indicates that the value of \( u \) remains unchanged regardless of how the parentheses are positioned.
(ii)Every monoid has a unique identity element.
(iii)Every element in a group has a unique inverse.
We prove the statement by induction on \( n \), assuming \( n \geq 3 \) Given a sequence \( u \) constructed from \( x_1, x_2, \ldots, x_n \), we can express \( u \) as \( u = v * w \), where \( v \) is formed from \( x_1, x_2, \ldots, x_i \) and \( w \) from \( x_{i+1}, \ldots, x_n \) for \( 1 \leq i \leq n-1 \) By induction, if \( i = n-1 \), then \( w = x_n \) and the result follows directly If \( i + 1 < n \), we define \( w = z * x_n \), where \( z \) is constructed from \( x_{i+1}, \ldots, x_{n-1} \) Thus, we can rewrite \( u \) as \( u = v * w = v * (z * x_n) = (v * z) * x_n \) using the associative law By induction, the result holds for \( v * z \), confirming it is also true for \( u \).
(ii) Suppose thateande are two identity elements in a monoid Thene=e∗e sincee is an identity; alsoe∗e =e sinceeis an identity Hencee=e
(iii) Letgbe an element of a group and supposeghas two inversesxandx ; we claim that x = x To see this observe that (x∗g)∗x = e∗x = x , while also
Due to the aforementioned points, we can confidently eliminate all parentheses from expressions created with elements \( x_1, x_2, \ldots, x_n \) in a semigroup, greatly enhancing simplicity Additionally, it is clear and unambiguous to refer to the identity element of a monoid and the inverse of an element within a group.
1 Let S be the subset of R ×R specified below and define (x, y)∗(x , y ) (x+x , y+y ) Say in each case whether(S,∗)is a semigroup, a monoid, a group, or none of these.
2 Do the sets of even or odd permutations inS n form a semigroup when functional composition is used as the binary operation?
3 Show that the set of all 2×2 real matrices with non-negative entries is a monoid but not a group when matrix addition used.
4 LetAbe a non-empty set and define a binary operation∗on the power set(P (A) byS∗T =(S∪T )−(S∩T ) Prove that(P (A),∗)is an abelian group.
5 Define powers in a semigroup(S,∗)by the rulesx 1 =xandx n + 1 =x n ∗xwhere x ∈Sandnis a non-negative integer Prove thatx m ∗ x n =x m + n and(x m ) n =x mn wherem, n >0.
6 LetGbe a monoid such that for eachxinGthere is a positive integernsuch that x n =e Prove thatGis a group.
Let G be the group formed by the permutations (1 2)(3 4), (1 3)(2 4), (1 4)(2 3), and the identity permutation (1)(2)(3)(4) This group contains exactly four elements, and each element serves as its own inverse Consequently, G is identified as the Klein 4-group, showcasing its unique properties in group theory.
8 Prove that the groupS n is abelian if and only ifn≤2.
9 Prove that the group GL n (R)is abelian if and only ifn=1.