DOCSLIB.ORG

MAT 102 Mathematical Systems & Groups Definition a Mathematical

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
MAT 102 Mathematical Systems & Groups Definition a Mathematical MAT 102 Mathematical Systems & Groups Definition A mathematical system (�, ∗) consists of a set � with elements �, �, �, �, … together with a binary operation ∗ that combines any two elements in � to create a new element (e.g. � ∗ �, � ∗ �, � ∗ �, etc.). Note that the set � can be either finite or infinite. Examples • (ℝ, ×) is the system of real numbers (ℝ) under multiplication. • (ℕ, −) is the system of counting numbers, or naturals, (ℕ) under subtraction. • The set of naturals 1 through 12 under “clock-12” addition or subtraction. These systems are generalized for any finite set (clocks with � hours) using mod � modular arithmetic. • The set {�, �} of odd and even integers under the usual operations of arithmetic. • The set {�, �} of truth values for statements under the “and” (conjunction) or “or (disjunction) connectives of propositional logic. • The set of symmetries of a square on a plane (i.e. the geometrical movements that keep the square unchanged) under rotations or reflections about an axis. Number Sets Below is a summary of some number sets commonly featured in mathematical systems. • ℙ = {2, 3, 5, 7, 11, 13, 17, 19, … } is the set of prime numbers. • ℕ = {1, 2, 3, 4, 5, … } is the set of naturals, or counting numbers. • � = {0, 1, 2, 3, 4, 5, … } is the set of whole numbers. • ℤ = {1, 2, 3, 4, 5, … } is the set of integers. • ℚ is the set of rational numbers, or fractions. These are the numbers whose decimal digits either terminate (e.g. 2⁄5 = 0.4) or have an unending but repeating pattern (e.g. 2⁄9 = 0.222 …). • � is the set of irrational numbers. These are the numbers whose decimal digits never terminate and do not have a repeating pattern (e.g. √2, �, 0.1011011101111…). • ℝ is the set of real numbers. These are all the rationals together with all the irrationals. They are the numbers that can be placed on a number line extending without bounds from left (−∞) to right (+∞). Properties Consider adding two numbers � and � from any of the number sets listed above. Many important properties turn out to be true for such a system. For example, the operation of addition is commutative since � + � = � + � and the number 0 acts as the additive identity since � + 0 = 0 + � = �. Other properties can also be verified for this system, such as associativity or the existence of inverses for � or � (provided the set of numbers is at least as big as the integers). Below are presented five important properties of mathematical systems. For all we consider the set of elements �, �, �, �, … of a set � under some binary operation ∗ that is well-defined. 1. Closure � is closed under the operation ∗ if, for any two elements �, � in �, the element � ∗ � is also in �. For example, the naturals are closed under addition since adding any two naturals always results in another, bigger, natural (e.g. 2 + 5 = 7). This same set, however, is not closed under subtraction since this operation can lead to negative integers (e.g. 2 − 5 = −3). Any instance, such as this last one, resulting in an element outside of � is called a counter-example. 2. Identity � has an identity element � if, for any element � in �, � ∗ � = � ∗ � = �. In other words, an identity is an element of � that, when combined through the operation ∗ with any other element in �, preserves that element. For example, 12 is the identity of the set of hours under “clock-12” (mod 12) addition since ℎ + 12 ≡ ℎ ��� 12 for ℎ = 1, 2, 3, … . , 12. In the usual systems of arithmetic, 0 is the additive identity and 1 is the multiplicative identity. 3. Inverses If an element � in � is such that � ∗ � = � ∗ � = �, then this element is called the inverse of � and (�, �) is called an inverse pair. For example, (7, 5) is an inverse pair of hours in the “clock-12” additive system since 5 + 7 ≡ 7 + 5 ≡ 12 ��� 12. With respect to the usual operations of arithmetic, the additive inverse of a number � is −� (the opposite of �) since � + (−�) = � − � = 0 and the multiplicative inverse of a non- S S T zero number � is (the reciprocal of �) since � × U V = = 1. Note that the number 0 T T T fails to have a multiplicative inverse since division by zero is undefined. 4. Associativity If for any three elements �, �, � in �, (� ∗ �) ∗ � = � ∗ (� ∗ �), the operation ∗ is said to be associative. This property is clearly true for addition and multiplication. For example, (1 + 2) + 3 = 3 + 3 = 6 and 1 + (2 + 3) = 1 + 5 = 6. However, the property fails for division or subtraction. For example, [(−2) − (−4)] − 5 = 2 − 5 = −3 yet (−2) − [(−4) − 5] = (−2) − (−9) = 7. 5. Commutativity If for any two elements �, � in �, � ∗ � = � ∗ �, the operation ∗ is said to be commutative. This property is also clearly true for addition and multiplication. For [ [ example, × 10 = 10 × = 4. Note again that this property fails for division or \ \ subtraction. For example, 3 ÷ 6 = 0.5 is different from 6 ÷ 3 = 2. Definition A mathematical system (�, ∗) that satisfies properties 1 through 4 is called a group. A group that also satisfies property 5 is called an Abelian (or commutative) group. What this definition implies then is that a group (�, ∗) must have a set � that is closed under ∗ and includes an identity element. Moreover, every element in � must have an inverse and the operation ∗ must be associative. Exercise 1 Is the set {1, 2, 3} under ��� 4 multiplication a group? Exercise 2 Is the set of truth values {�, �}, where � is “True” and � is “False”, for statements under the logical operation of conjunction a group? In logic, if � and � are basic statements that are either true (�) or false (�), then the conjunction of � and � is the compound statement “� and �.” According to the truth table for conjunction (see below), “� and �” is only true when both statements � and � are true. � � � and � T T T T F F F T F F F F Exercise 3 Are the following arithmetical systems groups? If yes, show that all four group properties hold for the system and check whether the group is Abelian (commutative). If no, show which of the four properties fail for the system. A) The set of whole numbers under subtraction. B) The set of positive rational numbers (i.e. fractions) under multiplication. C) The set of even integers under addition. D) The set of real numbers under division. Solution – Exercise 1 This system is not a group since it is not closed. As shown in the table below, 2 × 2 ≡ 0 ��� 4 and 0 is not in the set of numbers {1, 2, 3}. ��� � 1 2 3 multiplication 1 1 2 3 2 2 0 2 3 3 2 1 Checking all other properties, we find the following: • The set does have the identity 1 since any number 1, 2, or 3 multiplied by 1 mod 4 is itself (check this in the table with row 1 and column 1). • The numbers 1 and 3 have themselves as inverses, but 2 fails to have an inverse since the congruence 2 × � ≡ 1 ��� 4 has no solution � in the set {1, 2, 3}. This can be seen in the table by noting that 1 does not appear in the second row or the second column. • Associativity holds for this operation, though it is difficult to prove it. Here is an example that illustrates this property: (2 × 3) × 1 ≡ 2 × 1 ≡ 2 ��� 4 and 2 × (3 × 1) ≡ 2 × 3 ≡ 2 ��� 4. • Commutativity holds for this operation (e.g. 2 × 3 ≡ 3 × 2 ≡ 2 ��� 4) and it can be checked by noting that the numbers in the 3 × 3 table are symmetric with respect to the so- called main diagonal of entries (1, 0, 1) running from top-left to bottom-right. Solution – Exercise 2 The set of truth values under the logical operation of conjunction is not a group. As checked below, one of the elements in the set fails to have an inverse while all other group properties hold. § Closure The set is closed since the truth table for conjunction only contains the truth values T and F. § Identity The system has the identity T since T and T = T (row 1) and F and T = F (row 3). § Inverses Checking for inverses, we see that the inverse of T is T since T and T = T (row 1). However, F fails to have an inverse since a conjunction with a false component is never true! § Associativity It is easy to check that the logical operation of conjunction is associative. Solution – Exercise 3 A) The set of whole numbers under subtraction is not a group. As checked below, it violates all four group properties. § Closure The system is not closed. For example: 1 − 3 = −2, but −2 is not a whole number. § Identity & Inverses The system has no identity and, therefore, none of its elements have inverses. To see that there is no identity here, argue by contradiction. Assume � is the identity. Then, for any whole number �, we’d have � − � = � − � = �. But that would imply that � = 0 = �… Clearly this is not a possibility whenever � is positive. Therefore, there is no identity! § Associativity The operation of subtraction is not associative. In general, for any three whole numbers �, � and �, we have (� − �) − � = � − � − � ≠ � − (� − �) = � − � + �. For example, (5 − 3) − 1 = 2 − 1 = 1 ≠ 5 − (3 − 1) = 5 − 2 = 3. B) The set of positive fractions under multiplication is an Abelian group. As checked below, it satisfies all five properties for commutative groups. § Closure The system is closed since the product of any two positive fractions is also a positive fraction. § Identity T T T The system has the usual multiplicative identity 1 since × 1 = 1 × = for h h h T every positive fraction . h § Inverses T h T h Every positive fraction has its reciprocal as its inverse since × = 1.
Load more

Recommended publications
  • Algebra I (Math 200)
    Algebra I (Math 200) UCSC, Fall 2009 Robert Boltje Contents 1 Semigroups and Monoids 1 2 Groups 4 3 Normal Subgroups and Factor Groups 11 4 Normal and Subnormal Series 17 5 Group Actions 22 6 Symmetric and Alternating Groups 29 7 Direct and Semidirect Products 33 8 Free Groups and Presentations 35 9 Rings, Basic Definitions and Properties 40 10 Homomorphisms, Ideals and Factor Rings 45 11 Divisibility in Integral Domains 55 12 Unique Factorization Domains (UFD), Principal Ideal Do- mains (PID) and Euclidean Domains 60 13 Localization 65 14 Polynomial Rings 69 Chapter I: Groups 1 Semigroups and Monoids 1.1 Definition Let S be a set. (a) A binary operation on S is a map b : S × S ! S. Usually, b(x; y) is abbreviated by xy, x · y, x ∗ y, x • y, x ◦ y, x + y, etc. (b) Let (x; y) 7! x ∗ y be a binary operation on S. (i) ∗ is called associative, if (x ∗ y) ∗ z = x ∗ (y ∗ z) for all x; y; z 2 S. (ii) ∗ is called commutative, if x ∗ y = y ∗ x for all x; y 2 S. (iii) An element e 2 S is called a left (resp. right) identity, if e ∗ x = x (resp. x ∗ e = x) for all x 2 S. It is called an identity element if it is a left and right identity. (c) S together with a binary operation ∗ is called a semigroup, if ∗ is as- sociative. A semigroup (S; ∗) is called a monoid if it has an identity element. 1.2 Examples (a) Addition (resp.
    [Show full text]
  • Arithmetic Equivalence and Isospectrality
    ARITHMETIC EQUIVALENCE AND ISOSPECTRALITY ANDREW V.SUTHERLAND ABSTRACT. In these lecture notes we give an introduction to the theory of arithmetic equivalence, a notion originally introduced in a number theoretic setting to refer to number fields with the same zeta function. Gassmann established a direct relationship between arithmetic equivalence and a purely group theoretic notion of equivalence that has since been exploited in several other areas of mathematics, most notably in the spectral theory of Riemannian manifolds by Sunada. We will explicate these results and discuss some applications and generalizations. 1. AN INTRODUCTION TO ARITHMETIC EQUIVALENCE AND ISOSPECTRALITY Let K be a number field (a finite extension of Q), and let OK be its ring of integers (the integral closure of Z in K). The Dedekind zeta function of K is defined by the Dirichlet series X s Y s 1 ζK (s) := N(I)− = (1 N(p)− )− I OK p − ⊆ where the sum ranges over nonzero OK -ideals, the product ranges over nonzero prime ideals, and N(I) := [OK : I] is the absolute norm. For K = Q the Dedekind zeta function ζQ(s) is simply the : P s Riemann zeta function ζ(s) = n 1 n− . As with the Riemann zeta function, the Dirichlet series (and corresponding Euler product) defining≥ the Dedekind zeta function converges absolutely and uniformly to a nonzero holomorphic function on Re(s) > 1, and ζK (s) extends to a meromorphic function on C and satisfies a functional equation, as shown by Hecke [25]. The Dedekind zeta function encodes many features of the number field K: it has a simple pole at s = 1 whose residue is intimately related to several invariants of K, including its class number, and as with the Riemann zeta function, the zeros of ζK (s) are intimately related to the distribution of prime ideals in OK .
    [Show full text]
  • Boolean and Abstract Algebra Winter 2019
    Queen's University School of Computing CISC 203: Discrete Mathematics for Computing II Lecture 7: Boolean and Abstract Algebra Winter 2019 1 Boolean Algebras Recall from your study of set theory in CISC 102 that a set is a collection of items that are related in some way by a common property or rule. There are a number of operations that can be applied to sets, like [, \, and C. Combining these operations in a certain way allows us to develop a number of identities or laws relating to sets, and this is known as the algebra of sets. In a classical logic course, the first thing you typically learn about is propositional calculus, which is the branch of logic that studies propositions and connectives between propositions. For instance, \all men are mortal" and \Socrates is a man" are propositions, and using propositional calculus, we may conclude that \Socrates is mortal". In a sense, propositional calculus is very closely related to set theory, in that propo- sitional calculus is the study of the set of propositions together with connective operations on propositions. Moreover, we can use combinations of connective operations to develop the laws of propositional calculus as well as a collection of rules of inference, which gives us even more power to manipulate propositions. Before we continue, it is worth noting that the operations mentioned previously|and indeed, most of the operations we have been using throughout these notes|have a special name. Operations like [ and \ apply to pairs of sets in the same way that + and × apply to pairs of numbers.
    [Show full text]
  • 7.2 Binary Operators Closure
    last edited April 19, 2016 7.2 Binary Operators A precise discussion of symmetry benefits from the development of what math- ematicians call a group, which is a special kind of set we have not yet explicitly considered. However, before we define a group and explore its properties, we reconsider several familiar sets and some of their most basic features. Over the last several sections, we have considered many di↵erent kinds of sets. We have considered sets of integers (natural numbers, even numbers, odd numbers), sets of rational numbers, sets of vertices, edges, colors, polyhedra and many others. In many of these examples – though certainly not in all of them – we are familiar with rules that tell us how to combine two elements to form another element. For example, if we are dealing with the natural numbers, we might considered the rules of addition, or the rules of multiplication, both of which tell us how to take two elements of N and combine them to give us a (possibly distinct) third element. This motivates the following definition. Definition 26. Given a set S,abinary operator ? is a rule that takes two elements a, b S and manipulates them to give us a third, not necessarily distinct, element2 a?b. Although the term binary operator might be new to us, we are already familiar with many examples. As hinted to earlier, the rule for adding two numbers to give us a third number is a binary operator on the set of integers, or on the set of rational numbers, or on the set of real numbers.
    [Show full text]
  • 3. Closed Sets, Closures, and Density
    3. Closed sets, closures, and density 1 Motivation Up to this point, all we have done is define what topologies are, define a way of comparing two topologies, define a method for more easily specifying a topology (as a collection of sets generated by a basis), and investigated some simple properties of bases. At this point, we will start introducing some more interesting definitions and phenomena one might encounter in a topological space, starting with the notions of closed sets and closures. Thinking back to some of the motivational concepts from the first lecture, this section will start us on the road to exploring what it means for two sets to be \close" to one another, or what it means for a point to be \close" to a set. We will draw heavily on our intuition about n convergent sequences in R when discussing the basic definitions in this section, and so we begin by recalling that definition from calculus/analysis. 1 n Definition 1.1. A sequence fxngn=1 is said to converge to a point x 2 R if for every > 0 there is a number N 2 N such that xn 2 B(x) for all n > N. 1 Remark 1.2. It is common to refer to the portion of a sequence fxngn=1 after some index 1 N|that is, the sequence fxngn=N+1|as a tail of the sequence. In this language, one would phrase the above definition as \for every > 0 there is a tail of the sequence inside B(x)." n Given what we have established about the topological space Rusual and its standard basis of -balls, we can see that this is equivalent to saying that there is a tail of the sequence inside any open set containing x; this is because the collection of -balls forms a basis for the usual topology, and thus given any open set U containing x there is an such that x 2 B(x) ⊆ U.
    [Show full text]
  • Chapter 1 the Field of Reals and Beyond
    Chapter 1 The Field of Reals and Beyond Our goal with this section is to develop (review) the basic structure that character- izes the set of real numbers. Much of the material in the ¿rst section is a review of properties that were studied in MAT108 however, there are a few slight differ- ences in the de¿nitions for some of the terms. Rather than prove that we can get from the presentation given by the author of our MAT127A textbook to the previous set of properties, with one exception, we will base our discussion and derivations on the new set. As a general rule the de¿nitions offered in this set of Compan- ion Notes will be stated in symbolic form this is done to reinforce the language of mathematics and to give the statements in a form that clari¿es how one might prove satisfaction or lack of satisfaction of the properties. YOUR GLOSSARIES ALWAYS SHOULD CONTAIN THE (IN SYMBOLIC FORM) DEFINITION AS GIVEN IN OUR NOTES because that is the form that will be required for suc- cessful completion of literacy quizzes and exams where such statements may be requested. 1.1 Fields Recall the following DEFINITIONS: The Cartesian product of two sets A and B, denoted by A B,is a b : a + A F b + B . 1 2 CHAPTER 1. THE FIELD OF REALS AND BEYOND A function h from A into B is a subset of A B such that (i) 1a [a + A " 2bb + B F a b + h] i.e., dom h A,and (ii) 1a1b1c [a b + h F a c + h " b c] i.e., h is single-valued.
    [Show full text]
  • Real Numbers and Their Properties
    Real Numbers and their Properties Types of Numbers + • Z Natural numbers - counting numbers - 1, 2, 3,... The textbook uses the notation N. • Z Integers - 0, ±1, ±2, ±3,... The textbook uses the notation J. • Q Rationals - quotients (ratios) of integers. • R Reals - may be visualized as correspond- ing to all points on a number line. The reals which are not rational are called ir- rational. + Z ⊂ Z ⊂ Q ⊂ R. R ⊂ C, the field of complex numbers, but in this course we will only consider real numbers. Properties of Real Numbers There are four binary operations which take a pair of real numbers and result in another real number: Addition (+), Subtraction (−), Multiplication (× or ·), Division (÷ or /). These operations satisfy a number of rules. In the following, we assume a, b, c ∈ R. (In other words, a, b and c are all real numbers.) • Closure: a + b ∈ R, a · b ∈ R. This means we can add and multiply real num- bers. We can also subtract real numbers and we can divide as long as the denominator is not 0. • Commutative Law: a + b = b + a, a · b = b · a. This means when we add or multiply real num- bers, the order doesn’t matter. • Associative Law: (a + b) + c = a + (b + c), (a · b) · c = a · (b · c). We can thus write a + b + c or a · b · c without having to worry that different people will get different results. • Distributive Law: a · (b + c) = a · b + a · c, (a + b) · c = a · c + b · c. The distributive law is the one law which in- volves both addition and multiplication.
    [Show full text]
  • An Elementary Approach to Boolean Algebra
    Eastern Illinois University The Keep Plan B Papers Student Theses & Publications 6-1-1961 An Elementary Approach to Boolean Algebra Ruth Queary Follow this and additional works at: https://thekeep.eiu.edu/plan_b Recommended Citation Queary, Ruth, "An Elementary Approach to Boolean Algebra" (1961). Plan B Papers. 142. https://thekeep.eiu.edu/plan_b/142 This Dissertation/Thesis is brought to you for free and open access by the Student Theses & Publications at The Keep. It has been accepted for inclusion in Plan B Papers by an authorized administrator of The Keep. For more information, please contact [email protected]. r AN ELEr.:ENTARY APPRCACH TC BCCLF.AN ALGEBRA RUTH QUEAHY L _J AN ELE1~1ENTARY APPRCACH TC BC CLEAN ALGEBRA Submitted to the I<:athematics Department of EASTERN ILLINCIS UNIVERSITY as partial fulfillment for the degree of !•:ASTER CF SCIENCE IN EJUCATION. Date :---"'f~~-----/_,_ffo--..i.-/ _ RUTH QUEARY JUNE 1961 PURPOSE AND PLAN The purpose of this paper is to provide an elementary approach to Boolean algebra. It is designed to give an idea of what is meant by a Boclean algebra and to supply the necessary background material. The only prerequisite for this unit is one year of high school algebra and an open mind so that new concepts will be considered reason­ able even though they nay conflict with preconceived ideas. A mathematical science when put in final form consists of a set of undefined terms and unproved propositions called postulates, in terrrs of which all other concepts are defined, and from which all other propositions are proved.
    [Show full text]
  • Irreducible Representations of Finite Monoids
    U.U.D.M. Project Report 2019:11 Irreducible representations of finite monoids Christoffer Hindlycke Examensarbete i matematik, 30 hp Handledare: Volodymyr Mazorchuk Examinator: Denis Gaidashev Mars 2019 Department of Mathematics Uppsala University Irreducible representations of finite monoids Christoffer Hindlycke Contents Introduction 2 Theory 3 Finite monoids and their structure . .3 Introductory notions . .3 Cyclic semigroups . .6 Green’s relations . .7 von Neumann regularity . 10 The theory of an idempotent . 11 The five functors Inde, Coinde, Rese,Te and Ne ..................... 11 Idempotents and simple modules . 14 Irreducible representations of a finite monoid . 17 Monoid algebras . 17 Clifford-Munn-Ponizovski˘ıtheory . 20 Application 24 The symmetric inverse monoid . 24 Calculating the irreducible representations of I3 ........................ 25 Appendix: Prerequisite theory 37 Basic definitions . 37 Finite dimensional algebras . 41 Semisimple modules and algebras . 41 Indecomposable modules . 42 An introduction to idempotents . 42 1 Irreducible representations of finite monoids Christoffer Hindlycke Introduction This paper is a literature study of the 2016 book Representation Theory of Finite Monoids by Benjamin Steinberg [3]. As this book contains too much interesting material for a simple master thesis, we have narrowed our attention to chapters 1, 4 and 5. This thesis is divided into three main parts: Theory, Application and Appendix. Within the Theory chapter, we (as the name might suggest) develop the necessary theory to assist with finding irreducible representations of finite monoids. Finite monoids and their structure gives elementary definitions as regards to finite monoids, and expands on the basic theory of their structure. This part corresponds to chapter 1 in [3]. The theory of an idempotent develops just enough theory regarding idempotents to enable us to state a key result, from which the principal result later follows almost immediately.
    [Show full text]
  • Semigroups and Monoids 
    S Luis Alonso-Ovalle // Contents Subgroups Semigroups and monoids Subgroups Groups. A group G is an algebra consisting of a set G and a single binary operation ◦ satisfying the following axioms: . ◦ is completely defined and G is closed under ◦. ◦ is associative. G contains an identity element. Each element in G has an inverse element. Subgroups. We define a subgroup G0 as a subalgebra of G which is itself a group. Examples: . The group of even integers with addition is a proper subgroup of the group of all integers with addition. The group of all rotations of the square h{I, R, R0, R00}, ◦i, where ◦ is the composition of the operations is a subgroup of the group of all symmetries of the square. Some non-subgroups: SEMIGROUPS AND MONOIDS . The system h{I, R, R0}, ◦i is not a subgroup (and not even a subalgebra) of the original group. Why? (Hint: ◦ closure). The set of all non-negative integers with addition is a subalgebra of the group of all integers with addition, because the non-negative integers are closed under addition. But it is not a subgroup because it is not itself a group: it is associative and has a zero, but . does any member (except for ) have an inverse? Order. The order of any group G is the number of members in the set G. The order of any subgroup exactly divides the order of the parental group. E.g.: only subgroups of order , , and are possible for a -member group. (The theorem does not guarantee that every subset having the proper number of members will give rise to a subgroup.
    [Show full text]
  • Binary Operations
    Binary operations 1 Binary operations The essence of algebra is to combine two things and get a third. We make this into a definition: Definition 1.1. Let X be a set. A binary operation on X is a function F : X × X ! X. However, we don't write the value of the function on a pair (a; b) as F (a; b), but rather use some intermediate symbol to denote this value, such as a + b or a · b, often simply abbreviated as ab, or a ◦ b. For the moment, we will often use a ∗ b to denote an arbitrary binary operation. Definition 1.2. A binary structure (X; ∗) is a pair consisting of a set X and a binary operation on X. Example 1.3. The examples are almost too numerous to mention. For example, using +, we have (N; +), (Z; +), (Q; +), (R; +), (C; +), as well as n vector space and matrix examples such as (R ; +) or (Mn;m(R); +). Using n subtraction, we have (Z; −), (Q; −), (R; −), (C; −), (R ; −), (Mn;m(R); −), but not (N; −). For multiplication, we have (N; ·), (Z; ·), (Q; ·), (R; ·), (C; ·). If we define ∗ ∗ ∗ Q = fa 2 Q : a 6= 0g, R = fa 2 R : a 6= 0g, C = fa 2 C : a 6= 0g, ∗ ∗ ∗ then (Q ; ·), (R ; ·), (C ; ·) are also binary structures. But, for example, ∗ (Q ; +) is not a binary structure. Likewise, (U(1); ·) and (µn; ·) are binary structures. In addition there are matrix examples: (Mn(R); ·), (GLn(R); ·), (SLn(R); ·), (On; ·), (SOn; ·). Next, there are function composition examples: for a set X,(XX ; ◦) and (SX ; ◦).
    [Show full text]
  • Ring (Mathematics) 1 Ring (Mathematics)
    Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right.
    [Show full text]