DOCSLIB.ORG

Lecture 1. a Binary Operation Φ on a Set a Is A

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 1. a Binary Operation Φ on a Set a Is A Lecture 1. A binary operation φ on a set A is a function from A × A to A, i.e., φ : A × A → A. φ((x, y)) is also denoted by xφy. Often we use a symbol for φ: +, ·, − ,? . A field F is a set with two operation, called addition and multiplication, which are denoted by + and · (often omitted), respectively, and which satisfy the following properties: 1. Both operations are commutative: a + b = b + a and ab = ba for all a, b ∈ F 2. Both operations are associative:(a + b) + c = a + (b + c) and (ab)c = a(bc) for all a, b, c ∈ F 3. There exists a unique identity element for each operation, denoted by 0 and 1, i.e., 0 + a = a + 0 = a and 1a = a1 = a for all a ∈ F 4. For every a ∈ F, there exists a unique b ∈ F such that a + b = b + a = 0. This element b is called the additive inverse of a in F, and is denoted by −a. × 5. For every a ∈ F := F \{0}, there exists a unique b ∈ F such that ab = ba = 1. This −1 element b is called the multiplicative inverse of a in F, and is denoted by a . 6. Multiplication and addition are related by the distributive laws: a(b + c) = ab + ac and (b + c)a = ba + ca. The axiom system above is not the ‘most economic’ one. Check that it implies that 0a = a0 = 0 for every a ∈ F. Examples of fields. Q – the field of rational numbers; R – the field of real numbers; C – the field of complex numbers; Fp – the finite field of p elements, p is prime. The field Fp is often denoted by Z/pZ or Zp, and is thought as the set of p elements {0, 1, . , p − 1} where addition and multiplication are done by modulo p. Math 672: Lecture Notes. Fall 2005. Felix Lazebnik. 1 We have Q ⊂ R ⊂ C, and the operations in the subfields are just restrictions of the corresponding operations in C. Suppose V is a set whose elements are called vectors and denoted byv ¯. Suppose there exists a binary operation on V , called addition and denoted by +, which is commutative, associative, there exists an identity element, denoted by 0¯ and called zero vector, and for every vectorv ¯, there exists an additive inverse which we denote by −v¯. Suppose F is a field and there exists a function µ from µ : F × V → V given by (k, v¯) 7→ kv¯, which satisfies the following axioms: 1. 1v = v for allv ¯ ∈ V 2. k(¯v +u ¯) = kv¯ + ku¯ for all k ∈ F and allv, ¯ u¯ ∈ V 3. (k + m)¯v = kv¯ + mv¯ for all k ∈ F and allv ¯ ∈ V 4. k(mv¯) = (km)¯v for all k, m ∈ F and allv ¯ ∈ V An ordered triple ((V, +), F, µ) is called a vector space. In a simpler manner, we just say that V is a vector space over the field F. The function µ is mentioned very rarely. We say that µ defines a ‘multiplication’ of elements of F by vectors from V . Often in this context the elements of F are called scalars, and we say that µ defines a ‘multiplication of vectors by scalars’. Note that this multiplication is not a binary operation on a set. Its result is an element of V , i.e., is always a vector. When we write k(mv¯) = (km)¯v, we mean that the scalars k and m are multiplied in F, and the result of this multiplication km, which is a scalar, is ‘multiplied’ by a vectorv ¯. Examples of vector spaces. • For a field F, and positive integer n, let n V = F := {v¯ = (v1, v2, . , vn): vi ∈ F for all i = 1, 2, . , n} n The addition on F and the multiplication by scalars are defined as follows: for everyv, ¯ u¯ ∈ V , and every k ∈ F, v¯ +u ¯ := (v1 + u1, . , vn + un) and kv¯ = k(v1, . , vn) := (kv1, . , kvn). n Then V = F is a vector space over F, which can be (and has to be) checked easily. For F = R, n we obtain the well familiar space R . Math 672: Lecture Notes. Fall 2005. Felix Lazebnik. 2 n n 3 3 If F = Fp, vector space Fp contains p vectors. E.g., F2 contains 2 = 8 vectors. • Let V = C(0, 1) be the set of all continuous real valued functions on (0, 1): f : (0, 1) → R. The addition on C(0, 1) and the scalar multiplication are defined as follows: for any f, g ∈ C(0, 1), and every k ∈ R, (f + g)(x) := f(x) + g(x) and (kf)(x) := kf(x). Again, C(0, 1) is a vector space over R, which has to be checked. Here the fact that + is a binary operation on C(0, 1), or more precisely, that it is ‘closed’, which means f + g ∈ C(0, 1), is not a trivial matter. The same for kf, though it is a little easier. Similarly we can consider 1 C(R) or C (0, 1) – the vector space over R of all real valued differentiable functions from (0, 1) to R with continuous first derivative. • V = R is a vector space over Q. V = C is a vector space over R. V = C is a vector space over Q. In all these examples the addition in V is the usual addition in the field, and the multiplication of vectors by scalars is the usual multiplication of two numbers in V . • V = F[x] – set of all polynomials of x with coefficients from a field F. V is a vector space over F with respect to the usual addition of polynomials and the multiplication of polynomials by numbers (elements of F). • V is the set of all functions f : R → R which satisfy the differential equation y00 − 5y0 + 6y = 0. V is a vector space over R. • V is the set of all sequences of real numbers (xn), n ≥ 0, defined by the recurrences: x0 = a, x1 = b, a, b ∈ R, and for all n ≥ 0, xn+2 = 2xn+1 − 3xn. V is a vector space over R. • V = Mm×n(F) – the set of all m × n matrices with entries from F with respect to usual addition of matrices and the multiplication of matrices by scalars. • Let A ∈ Mm×n, i.e., A is an m × n matrix over R. Then the set V of all solutions of the n homogeneous system of linear equations Ax¯ = 0,¯ i.e., the set of all vectorsx ¯ ∈ R such that Ax¯ = 0,¯ is a vector space over R with respect to usual addition of vectors and the scalar n n multiplication in R . Note that in this example elements of R are thought of as the column vectors ( n × 1 matrices). Proposition 1 Let V be a vector space over a field F. Then Math 672: Lecture Notes. Fall 2005. Felix Lazebnik. 3 (i) 0¯ is unique. (ii) for each a¯ ∈ V , the inverse of a¯ is unique. (iii) 0¯a = 0¯ for every a¯ ∈ V . (iv) (−1)¯a = −a¯ for every a¯ ∈ V . (v) −(−a¯) =a ¯ for every a¯ ∈ V . (vi) Cancellation Law: a¯ + ¯b =a ¯ +c ¯ if and only if ¯b =c ¯. Proof. (i) Indeed, if 0¯0 is a possible another identity element, then 0¯ + 0¯0 = 0¯ as 0¯0 is an identity, and 0¯ + 0¯0 = 0¯0 as 0¯ is an identity. So 0¯ = 0¯0. This justifies the notation 0.¯ (ii) Indeed, let ¯b, ¯b0 be inverses of a. Then consider the element (¯b +a ¯) + ¯b0. Since ¯b +a ¯ = 0,¯ then (¯b +a ¯) + ¯b0 = 0¯ + ¯b0 = ¯b0. Similarly, consider ¯b + (¯a + ¯b0). Sincea ¯ + ¯b0 = 0,¯ then ¯b + (¯a + ¯b0) = ¯b + 0¯ = ¯b. Due to the associativity, (¯b +a ¯) + ¯b0 = ¯b + (¯a + ¯b0). So ¯b0 = ¯b. This justifies the notation −a¯. (iii) (iv) (v) (vi) Math 672: Lecture Notes. Fall 2005. Felix Lazebnik. 4 From now on: We will NOT denote vectors by bars. When we write kv, we will mean that k ∈ F and v ∈ V . If we do not say otherwise , V will denote a vector space over an arbitrary field F. Math 672: Lecture Notes. Fall 2005. Felix Lazebnik. 5 Lecture 2. Let V be a vector space and k1, . , km ∈ F and v1, . , vm ∈ V . Then the vector k1v1 + ... + kmvm is called a linear combination of v1, . , vm. At this time we do not define infinite linear combinations. For a subset A of V the set of all (finite) linear combinations of vectors from A is called the span of A and is denoted by Span(A) or hAi. Clearly, A ⊆ Span(A). P Sometimes it is convenient to use a∈A kaa as a notation for a general linear combination of finitely many elements of A. In this notation we always assume that only finitely many coefficients ka are nonzero A subset W of a vector space V over a field F is called a subspace or a linear subspace of V if it is a vector space over the the operation on V restricted to W and the multiplication of elements of V by elements from F restricted to W . In order to check that a subset W is a subspace in V it is sufficient to check that it is “closed” with respect to the addition of V and with respect to the multiplication by scalars. All other axioms will be inherited from the ones in V . The existence of the zero vector in W and the additive inverses follows from the fact that in V 0a = 0 (= 0),¯ (−1)a = −a and that W is closed with respect to multiplication of vectors by scalars.
Load more

Recommended publications
  • 2.1 the Algebra of Sets
    Chapter 2 I Abstract Algebra 83 part of abstract algebra, sets are fundamental to all areas of mathematics and we need to establish a precise language for sets. We also explore operations on sets and relations between sets, developing an “algebra of sets” that strongly resembles aspects of the algebra of sentential logic. In addition, as we discussed in chapter 1, a fundamental goal in mathematics is crafting articulate, thorough, convincing, and insightful arguments for the truth of mathematical statements. We continue the development of theorem-proving and proof-writing skills in the context of basic set theory. After exploring the algebra of sets, we study two number systems denoted Zn and U(n) that are closely related to the integers. Our approach is based on a widely used strategy of mathematicians: we work with specific examples and look for general patterns. This study leads to the definition of modified addition and multiplication operations on certain finite subsets of the integers. We isolate key axioms, or properties, that are satisfied by these and many other number systems and then examine number systems that share the “group” properties of the integers. Finally, we consider an application of this mathematics to check digit schemes, which have become increasingly important for the success of business and telecommunications in our technologically based society. Through the study of these topics, we engage in a thorough introduction to abstract algebra from the perspective of the mathematician— working with specific examples to identify key abstract properties common to diverse and interesting mathematical systems. 2.1 The Algebra of Sets Intuitively, a set is a “collection” of objects known as “elements.” But in the early 1900’s, a radical transformation occurred in mathematicians’ understanding of sets when the British philosopher Bertrand Russell identified a fundamental paradox inherent in this intuitive notion of a set (this paradox is discussed in exercises 66–70 at the end of this section).
    [Show full text]
  • Mathematics I Basic Mathematics
    Mathematics I Basic Mathematics Prepared by Prof. Jairus. Khalagai African Virtual university Université Virtuelle Africaine Universidade Virtual Africana African Virtual University NOTICE This document is published under the conditions of the Creative Commons http://en.wikipedia.org/wiki/Creative_Commons Attribution http://creativecommons.org/licenses/by/2.5/ License (abbreviated “cc-by”), Version 2.5. African Virtual University Table of ConTenTs I. Mathematics 1, Basic Mathematics _____________________________ 3 II. Prerequisite Course or Knowledge _____________________________ 3 III. Time ____________________________________________________ 3 IV. Materials _________________________________________________ 3 V. Module Rationale __________________________________________ 4 VI. Content __________________________________________________ 5 6.1 Overview ____________________________________________ 5 6.2 Outline _____________________________________________ 6 VII. General Objective(s) ________________________________________ 8 VIII. Specific Learning Objectives __________________________________ 8 IX. Teaching and Learning Activities ______________________________ 10 X. Key Concepts (Glossary) ____________________________________ 16 XI. Compulsory Readings ______________________________________ 18 XII. Compulsory Resources _____________________________________ 19 XIII. Useful Links _____________________________________________ 20 XIV. Learning Activities _________________________________________ 23 XV. Synthesis Of The Module ___________________________________
    [Show full text]
  • The Five Fundamental Operations of Mathematics: Addition, Subtraction
    The five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms Kenneth A. Ribet UC Berkeley Trinity University March 31, 2008 Kenneth A. Ribet Five fundamental operations This talk is about counting, and it’s about solving equations. Counting is a very familiar activity in mathematics. Many universities teach sophomore-level courses on discrete mathematics that turn out to be mostly about counting. For example, we ask our students to find the number of different ways of constituting a bag of a dozen lollipops if there are 5 different flavors. (The answer is 1820, I think.) Kenneth A. Ribet Five fundamental operations Solving equations is even more of a flagship activity for mathematicians. At a mathematics conference at Sundance, Robert Redford told a group of my colleagues “I hope you solve all your equations”! The kind of equations that I like to solve are Diophantine equations. Diophantus of Alexandria (third century AD) was Robert Redford’s kind of mathematician. This “father of algebra” focused on the solution to algebraic equations, especially in contexts where the solutions are constrained to be whole numbers or fractions. Kenneth A. Ribet Five fundamental operations Here’s a typical example. Consider the equation y 2 = x3 + 1. In an algebra or high school class, we might graph this equation in the plane; there’s little challenge. But what if we ask for solutions in integers (i.e., whole numbers)? It is relatively easy to discover the solutions (0; ±1), (−1; 0) and (2; ±3), and Diophantus might have asked if there are any more.
    [Show full text]
  • A Quick Algebra Review
    A Quick Algebra Review 1. Simplifying Expressions 2. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Exponents 9. Quadratics 10. Rationals 11. Radicals Simplifying Expressions An expression is a mathematical “phrase.” Expressions contain numbers and variables, but not an equal sign. An equation has an “equal” sign. For example: Expression: Equation: 5 + 3 5 + 3 = 8 x + 3 x + 3 = 8 (x + 4)(x – 2) (x + 4)(x – 2) = 10 x² + 5x + 6 x² + 5x + 6 = 0 x – 8 x – 8 > 3 When we simplify an expression, we work until there are as few terms as possible. This process makes the expression easier to use, (that’s why it’s called “simplify”). The first thing we want to do when simplifying an expression is to combine like terms. For example: There are many terms to look at! Let’s start with x². There Simplify: are no other terms with x² in them, so we move on. 10x x² + 10x – 6 – 5x + 4 and 5x are like terms, so we add their coefficients = x² + 5x – 6 + 4 together. 10 + (-5) = 5, so we write 5x. -6 and 4 are also = x² + 5x – 2 like terms, so we can combine them to get -2. Isn’t the simplified expression much nicer? Now you try: x² + 5x + 3x² + x³ - 5 + 3 [You should get x³ + 4x² + 5x – 2] Order of Operations PEMDAS – Please Excuse My Dear Aunt Sally, remember that from Algebra class? It tells the order in which we can complete operations when solving an equation.
    [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]
  • Axioms and Algebraic Systems*
    Axioms and algebraic systems* Leong Yu Kiang Department of Mathematics National University of Singapore In this talk, we introduce the important concept of a group, mention some equivalent sets of axioms for groups, and point out the relationship between the individual axioms. We also mention briefly the definitions of a ring and a field. Definition 1. A binary operation on a non-empty set S is a rule which associates to each ordered pair (a, b) of elements of S a unique element, denoted by a* b, in S. The binary relation itself is often denoted by *· It may also be considered as a mapping from S x S to S, i.e., * : S X S ~ S, where (a, b) ~a* b, a, bE S. Example 1. Ordinary addition and multiplication of real numbers are binary operations on the set IR of real numbers. We write a+ b, a· b respectively. Ordinary division -;- is a binary relation on the set IR* of non-zero real numbers. We write a -;- b. Definition 2. A binary relation * on S is associative if for every a, b, c in s, (a* b) * c =a* (b *c). Example 2. The binary operations + and · on IR (Example 1) are as­ sociative. The binary relation -;- on IR* (Example 1) is not associative smce 1 3 1 ~ 2) ~ 3 _J_ - 1 ~ (2 ~ 3). ( • • = -6 I 2 = • • * Talk given at the Workshop on Algebraic Structures organized by the Singapore Mathemat- ical Society for school teachers on 5 September 1988. 11 Definition 3. A semi-group is a non-€mpty set S together with an asso­ ciative binary operation *, and is denoted by (S, *).
    [Show full text]
  • Rules for Matrix Operations
    Math 2270 - Lecture 8: Rules for Matrix Operations Dylan Zwick Fall 2012 This lecture covers section 2.4 of the textbook. 1 Matrix Basix Most of this lecture is about formalizing rules and operations that we’ve already been using in the class up to this point. So, it should be mostly a review, but a necessary one. If any of this is new to you please make sure you understand it, as it is the foundation for everything else we’ll be doing in this course! A matrix is a rectangular array of numbers, and an “m by n” matrix, also written rn x n, has rn rows and n columns. We can add two matrices if they are the same shape and size. Addition is termwise. We can also mul tiply any matrix A by a constant c, and this multiplication just multiplies every entry of A by c. For example: /2 3\ /3 5\ /5 8 (34 )+( 10 Hf \i 2) \\2 3) \\3 5 /1 2\ /3 6 3 3 ‘ = 9 12 1 I 1 2 4) \6 12 1 Moving on. Matrix multiplication is more tricky than matrix addition, because it isn’t done termwise. In fact, if two matrices have the same size and shape, it’s not necessarily true that you can multiply them. In fact, it’s only true if that shape is square. In order to multiply two matrices A and B to get AB the number of columns of A must equal the number of rows of B. So, we could not, for example, multiply a 2 x 3 matrix by a 2 x 3 matrix.
    [Show full text]
  • A List of Arithmetical Structures Complete with Respect to the First
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Theoretical Computer Science 257 (2001) 115–151 www.elsevier.com/locate/tcs A list of arithmetical structures complete with respect to the ÿrst-order deÿnability Ivan Korec∗;X Slovak Academy of Sciences, Mathematical Institute, Stefanikovaà 49, 814 73 Bratislava, Slovak Republic Abstract A structure with base set N is complete with respect to the ÿrst-order deÿnability in the class of arithmetical structures if and only if the operations +; × are deÿnable in it. A list of such structures is presented. Although structures with Pascal’s triangles modulo n are preferred a little, an e,ort was made to collect as many simply formulated results as possible. c 2001 Elsevier Science B.V. All rights reserved. MSC: primary 03B10; 03C07; secondary 11B65; 11U07 Keywords: Elementary deÿnability; Pascal’s triangle modulo n; Arithmetical structures; Undecid- able theories 1. Introduction A list of (arithmetical) structures complete with respect of the ÿrst-order deÿnability power (shortly: def-complete structures) will be presented. (The term “def-strongest” was used in the previous versions.) Most of them have the base set N but also structures with some other universes are considered. (Formal deÿnitions are given below.) The class of arithmetical structures can be quasi-ordered by ÿrst-order deÿnability power. After the usual factorization we obtain a partially ordered set, and def-complete struc- tures will form its greatest element. Of course, there are stronger structures (with respect to the ÿrst-order deÿnability) outside of the class of arithmetical structures.
    [Show full text]
  • Right Ideals of a Ring and Sublanguages of Science
    RIGHT IDEALS OF A RING AND SUBLANGUAGES OF SCIENCE Javier Arias Navarro Ph.D. In General Linguistics and Spanish Language http://www.javierarias.info/ Abstract Among Zellig Harris’s numerous contributions to linguistics his theory of the sublanguages of science probably ranks among the most underrated. However, not only has this theory led to some exhaustive and meaningful applications in the study of the grammar of immunology language and its changes over time, but it also illustrates the nature of mathematical relations between chunks or subsets of a grammar and the language as a whole. This becomes most clear when dealing with the connection between metalanguage and language, as well as when reflecting on operators. This paper tries to justify the claim that the sublanguages of science stand in a particular algebraic relation to the rest of the language they are embedded in, namely, that of right ideals in a ring. Keywords: Zellig Sabbetai Harris, Information Structure of Language, Sublanguages of Science, Ideal Numbers, Ernst Kummer, Ideals, Richard Dedekind, Ring Theory, Right Ideals, Emmy Noether, Order Theory, Marshall Harvey Stone. §1. Preliminary Word In recent work (Arias 2015)1 a line of research has been outlined in which the basic tenets underpinning the algebraic treatment of language are explored. The claim was there made that the concept of ideal in a ring could account for the structure of so- called sublanguages of science in a very precise way. The present text is based on that work, by exploring in some detail the consequences of such statement. §2. Introduction Zellig Harris (1909-1992) contributions to the field of linguistics were manifold and in many respects of utmost significance.
    [Show full text]
  • Basic Concepts of Set Theory, Functions and Relations 1. Basic
    Ling 310, adapted from UMass Ling 409, Partee lecture notes March 1, 2006 p. 1 Basic Concepts of Set Theory, Functions and Relations 1. Basic Concepts of Set Theory........................................................................................................................1 1.1. Sets and elements ...................................................................................................................................1 1.2. Specification of sets ...............................................................................................................................2 1.3. Identity and cardinality ..........................................................................................................................3 1.4. Subsets ...................................................................................................................................................4 1.5. Power sets .............................................................................................................................................4 1.6. Operations on sets: union, intersection...................................................................................................4 1.7 More operations on sets: difference, complement...................................................................................5 1.8. Set-theoretic equalities ...........................................................................................................................5 Chapter 2. Relations and Functions ..................................................................................................................6
    [Show full text]
  • Formal Introduction to Digital Image Processing
    MINISTÉRIO DA CIÊNCIA E TECNOLOGIA INSTITUTO NACIONAL DE PESQUISAS ESPACIAIS INPE–7682–PUD/43 Formal introduction to digital image processing Gerald Jean Francis Banon INPE São José dos Campos July 2000 Preface The objects like digital image, scanner, display, look–up–table, filter that we deal with in digital image processing can be defined in a precise way by using various algebraic concepts such as set, Cartesian product, binary relation, mapping, composition, opera- tion, operator and so on. The useful operations on digital images can be defined in terms of mathematical prop- erties like commutativity, associativity or distributivity, leading to well known algebraic structures like monoid, vector space or lattice. Furthermore, the useful transformations that we need to process the images can be defined in terms of mappings which preserve these algebraic structures. In this sense, they are called morphisms. The main objective of this book is to give all the basic details about the algebraic ap- proach of digital image processing and to cover in a unified way the linear and morpholog- ical aspects. With all the early definitions at hand, apparently difficult issues become accessible. Our feeling is that such a formal approach can help to build a unified theory of image processing which can benefit the specification task of image processing systems. The ultimate goal would be a precise characterization of any research contribution in this area. This book is the result of many years of works and lectures in signal processing and more specifically in digital image processing and mathematical morphology. Within this process, the years we have spent at the Brazilian Institute for Space Research (INPE) have been decisive.
    [Show full text]
  • Binary Operations
    4-22-2007 Binary Operations Definition. A binary operation on a set X is a function f : X × X → X. In other words, a binary operation takes a pair of elements of X and produces an element of X. It’s customary to use infix notation for binary operations. Thus, rather than write f(a, b) for the binary operation acting on elements a, b ∈ X, you write afb. Since all those letters can get confusing, it’s also customary to use certain symbols — +, ·, ∗ — for binary operations. Thus, f(a, b) becomes (say) a + b or a · b or a ∗ b. Example. Addition is a binary operation on the set Z of integers: For every pair of integers m, n, there corresponds an integer m + n. Multiplication is also a binary operation on the set Z of integers: For every pair of integers m, n, there corresponds an integer m · n. However, division is not a binary operation on the set Z of integers. For example, if I take the pair 3 (3, 0), I can’t perform the operation . A binary operation on a set must be defined for all pairs of elements 0 from the set. Likewise, a ∗ b = (a random number bigger than a or b) does not define a binary operation on Z. In this case, I don’t have a function Z × Z → Z, since the output is ambiguously defined. (Is 3 ∗ 5 equal to 6? Or is it 117?) When a binary operation occurs in mathematics, it usually has properties that make it useful in con- structing abstract structures.
    [Show full text]