DOCSLIB.ORG

Multiplication Operation on Fuzzy Numbers

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Multiplication Operation on Fuzzy Numbers JOURNAL OF SOFTWARE, VOL. 4, NO. 4, JUNE 2009 331 Multiplication Operation on Fuzzy Numbers Shang Gao and Zaiyue Zhang School of Computer Science and Engineering, Jiangsu University of Science and Technology, Zhenjiang 212003, China Email: [email protected] [email protected] Cungen Cao Institute of Computing Technology,The Chinese Academy of Sciences,Beijing 100080, China Email: [email protected] Abstract—A fuzzy number is simply an ordinary number programming, engineering (especially communications), whose precise value is somewhat uncertain. Fuzzy numbers and experimental science. S. M. Wang and J. Watada [6] are used in statistics, computer programming, engineering, discuss the laws of large numbers for T-independent L-R and experimental science. The arithmetic operators on fuzzy fuzzy variables based on continuous archimedean t-norm numbers are basic content in fuzzy mathematics. Operation and expected value of fuzzy variable. G. X. Wang, Q. of fuzzy number can be generalized from that of crisp interval. The operations of interval are discussed. Zhang and X. J. Cui [8] define a special sort discrete Multiplication operation on fuzzy numbers is defined by the fuzzy numbers - discrete fuzzy number on a fixed set extension principle. Based on extension principle, nonlinear with finite support set, and then obtain a representation programming method, analytical method, computer theorem of such discrete fuzzy numbers, study the drawing method and computer simulation method are used operations of scalar product, addition and multiplication, for solving multiplication operation of two fuzzy numbers. and obtain some results. H. M. Lee and L. Lin [9] The nonlinear programming method is a precise method weighted triangular fuzzy numbers to tackle the rate of also, but it only gets a membership value as given number aggregative risk in fuzzy circumstances during any phase and it is a difficult problem for solving nonlinear of the software development life cycle. X. W. Zhou, L. P. programming. The analytical method is most precise, but it is hard to α -cuts interval when the membership function is Wang and B. H. Zheng [10] calculated concentration of complicated. The computer drawing method is simple, but it pollution by means of triangle fuzzy number and need calculate the α -cuts interval. The computer established fuzzy risk assessment model of the potential simulation method is the most simple, and it has wide ecological risk index. D. Sanchez, M. Delgado and M. A. applicability, but the membership function is rough. Each Vila [11] define imprecise quantities on the basis of a method is illuminated by examples. new representation of imprecision introduced by the authors called RL-representation and show that the Index Terms—fuzzy number, membership function, imprecision of the quantities being operated can be extension principle, α -cuts; nonlinear programming increased, preserved or diminished. The concept takes into account the fact that all phenomena in the physical universe have a degree of I. INTRODUCTION inherent uncertainty. The arithmetic operators on fuzzy In most of cases in our life, the data obtained for numbers are basic content in fuzzy mathematics. decision making are only approximately known. In1965, Multiplication operation on fuzzy numbers is defined by Zadeh [1] introduced the concept of fuzzy set theory to the extension principle. The procedure of addition or meet those problems. In 1978, Dubois and Prade defined subtraction is simple, but the procedure of multiplication any of the fuzzy numbers as a fuzzy subset of the real or division is complex. The nonlinear programming, line [2]. Fuzzy numbers allow us to make the analytical method, computer drawing and computer mathematical model of linguistic variable or fuzzy simulation method are used for solving multiplication environment. A fuzzy number is a quantity whose value operation of two fuzzy numbers. The procedure of is imprecise, rather than exact as is the case with division is similar. "ordinary" (single-valued) numbers [3-6]. Any fuzzy number can be thought of as a function whose domain is II. CONCEPT OF FUZZY NUMBER a specified set. In many respects, fuzzy numbers depict the physical world more realistically than single-valued A. Fuzzy Number numbers. Fuzzy numbers are used in statistics, computer If a fuzzy set is convex and normalized, and its membership function is defined in R and piecewise continuous, it is called as fuzzy number. So fuzzy number Supported by the National Natural Science Foundation of China under Grant No.60773059. © 2009 ACADEMY PUBLISHER 332 JOURNAL OF SOFTWARE, VOL. 4, NO. 4, JUNE 2009 (fuzzy set) represents a real number interval whose (α'< α) ⇒ (a (α ') ≤ a (α ) ,a (α ') ≥ a (α ) ) boundary is fuzzy. 1 1 3 3 Fuzzy number is expressed as a fuzzy set defining a The convex condition may also be written as, fuzzy interval in the real number R. Since the boundary of this interval is ambiguous, the interval is also a fuzzy (α'< α) ⇒ (A ⊂ A ) set. Generally a fuzzy interval is represented by two end α α ' Operation of fuzzy number can be generalized from points a1 and a3 and a peak point a2 as [a1, a2, a3 ] (Figure 1). The α -cut operation can be also applied to the fuzzy that of crisp interval. Let’s have a look at the operations of interval. number. If we denote α -cut interval for fuzzy number A as Aα , the obtained interval Aα is defined as ∀ ∈ a1, a3, b1, b3 R (α ) (α ) Aα = [a1 ,a3 ] A = [a1, a3], B = [b1, b3] We can also know that it is an ordinary crisp interval Assuming A and B as numbers expressed as interval, (Figure 2). main operations of interval are i) Addition [a1, a3] (+) [b1, b3] = [a1 + b1, a3 + b3] ii) Subtraction [a1, a3] (-) [b1, b3] = [a1 - b3, a3- b1] iii) Multiplication ∧ ∧ ∧ [a1, a3] (•) [b1, b3] = [a1 • b1 a1 • b3 a3 • b1 a3 • b3, a • b ∨ a • b ∨ a • b ∨ a • b ] Figure 1 Fuzzy Number A = [a1 ,a2 ,a3 ] 1 1 1 3 3 1 3 3 Fuzzy number should be normalized and convex. Here iv) Division the condition of normalization implies that maximum membership value is 1. ∧ ∧ ∧ [a1, a3] (/) [b1, b3] = [a1 / b1 a1 / b3 a3 / b1 a3 / b3, ∃x ∈ R , µ ~ (x ) = 1 ∨ ∨ ∨ 0 A 0 a1 / b1 a1 / b3 a3 / b1 a3 / b3] B. Operation of α -cut Interval excluding the case b1 = 0 or b3 = 0 v) Inverse interval -1 ∧ ∨ [a1, a3] = [1 / a1 1 / a3, 1 / a1 1 / a3] excluding the case a1 = 0 or a3 = 0 When previous sets A and B is defined in the positive real number R+, the operations of multiplication, division, and inverse interval are written as, iii′) Multiplication [a , a ] (•) [b , b ] = [a • b , a • b ] 1 3 1 3 1 1 3 3 Figure 2 α -cut of fuzzy number iv′) Division (α'< α) ⇒ (Aα ⊂ Aα ' ) [a , a ] (/) [b , b ] = [a / b , a / b ] The convex condition is that the line by α -cut is 1 3 1 3 1 3 3 1 continuous and α -cut interval satisfies the following v) Inverse Interval relation. -1 (α ) (α ) [a1, a3] = [1 / a3, 1 / a1] Aα = [a1 ,a3 ] © 2009 ACADEMY PUBLISHER JOURNAL OF SOFTWARE, VOL. 4, NO. 4, JUNE 2009 333 vi) Minimum Therefore multiplication operation on fuzzy numbers is expressed as [a , a ] (∧) [b , b ] = [a ∧ b , a ∧ b ] 1 3 1 3 1 1 3 3 ~ ~ ~ ~ µ M (×)N (z) = sup min{µ M (x),µ N (y)} (2) z=x×y vii) Maximum The procedure of addition or subtraction is simple, ∨ ∨ ∨ but the procedure of multiplication or division is complex. [a1, a3] ( ) [b1, b3] = [a1 b1, a3 b3] Example 1 There are two intervals A and B, III. NONLINEAR PROGRAMMING METHOD Based on multiplication operation on fuzzy numbers, A = [3, 5], B = [-2, 7] multiplication operation problem is formulated as a nonlinear programming. Then following operation might be set. max v A(+)B=[3-2, 5+7]=[1, 12] ~ s.t. µ M (x) ≥ v (3) A(-)B=[3-7, 5-(-2)]=[-4, 7] ~ µ N (y) ≥ v ∧ ∧ ∧ A(•)B=[3•(-2) 3•7 5•(-2) 5•7, xy = z0 3•(-2) ∨3•7∨5•(-2) ∨5•7] Given z0 , we can get the maximum vmax of nonlinear programming (1). vmax is membership value of z0 . =[-10, 35] ~ ~ Example 2 Suppose that the membership of M , N A(/)B=[3/(-2) ∧3/7∧5/(-2) ∧5/7, −(x−2)2 are ~ (Figure 3) and µ M (x) = e 3/(-2) ∨3/7∨5/(-2) ∨5/7] −( y+2)2 ~ ~ ~ ~ (Figure 4). is µ N (y) = e Q = M (×)N =[-2.5, 5/7] formulated as following: -1 -1 max v A =[3,5] = [1/5, 1/3] 2 s.t. e −(x−2) ≥ v -1 -1 ∧ ∨ 2 (4) B =[-2,7] = [1/(-2) 1/7, 1/(-2) 1/7] e −( y+2) ≥ v =[-1/2, 1/7] xy = z0 A(∧)B=[3∧(-2),5∧7] When z0 =-12,-11,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1 =[-2, 5] respectively, we can get the vmax correspondingly. The membership value of z is shown in table 1 and A(∨)B=[3∨(-2),5∨7] 0 membership function is shown in Figure 5. =[3, 7] C. Operation of fuzzy numbers Based on the extension principle, arithmetic operations on fuzzy numbers are defined by following: ~ ~ If M and N are fuzzy numbers, membership of ~ ~ M (∗)N is defined as follow: µ ~ ~ (z) = sup min µ ~ (x),µ ~ (y) (1) ~ M (∗)N { M N } Figure 3 Membership function of z=x∗y M Where * stands for any of the four arithmetic operations. ~ ~ ~ ~ µ M (+)N (z) = sup min{µ M (x),µ N (y)} z=x+ y ~ ~ ~ ~ µ M (−)N (z) = sup min{µ M (x),µ N (y)} z=x− y ~ ~ ~ ~ µ M (/)N (z) = sup min{µ M (x),µ N (y)} z=x / y ~ Figure 4 Membership function of N © 2009 ACADEMY PUBLISHER 334 JOURNAL OF SOFTWARE, VOL.
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]
  • 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]
  • 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]
  • 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]
  • 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]
  • 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]
  • 1.3 Matrices and Matrix Operations 1.3.1 De…Nitions and Notation Matrices Are Yet Another Mathematical Object
    20 CHAPTER 1. SYSTEMS OF LINEAR EQUATIONS AND MATRICES 1.3 Matrices and Matrix Operations 1.3.1 De…nitions and Notation Matrices are yet another mathematical object. Learning about matrices means learning what they are, how they are represented, the types of operations which can be performed on them, their properties and …nally their applications. De…nition 50 (matrix) 1. A matrix is a rectangular array of numbers. in which not only the value of the number is important but also its position in the array. 2. The numbers in the array are called the entries of the matrix. 3. The size of the matrix is described by the number of its rows and columns (always in this order). An m n matrix is a matrix which has m rows and n columns. 4. The elements (or the entries) of a matrix are generally enclosed in brack- ets, double-subscripting is used to index the elements. The …rst subscript always denote the row position, the second denotes the column position. For example a11 a12 ::: a1n a21 a22 ::: a2n A = 2 ::: ::: ::: ::: 3 (1.5) 6 ::: ::: ::: ::: 7 6 7 6 am1 am2 ::: amn 7 6 7 =4 [aij] , i = 1; 2; :::; m, j =5 1; 2; :::; n (1.6) Enclosing the general element aij in square brackets is another way of representing a matrix A . 5. When m = n , the matrix is said to be a square matrix. 6. The main diagonal in a square matrix contains the elements a11; a22; a33; ::: 7. A matrix is said to be upper triangular if all its entries below the main diagonal are 0.
    [Show full text]
  • Operation CLUE WORDS Remember, Read Each Question Carefully
    Operation CLUE WORDS Remember, read each question carefully. THINK about what the question is asking. Addition Subtraction add difference altogether fewer and gave away both take away in all how many more sum how much longer/ total shorter/smaller increase left less change Multiplication decrease each same twice Division product share equally in all (each) each double quotient every ©2012 Amber Polk-Adventures of a Third Grade Teacher Clue Word Cut and Paste Cut out the clue words. Sort and glue under the operation heading. Subtraction Multiplication Addition Division add each difference share equally both fewer in all twice product every gave away change quotient sum total left double less decrease increase ©2012 Amber Polk-Adventures of a Third Grade Teacher Clue Word Problems Underline the clue word in each word problem. Write the equation. Solve the problem. 1. Mark has 17 gum balls. He 5. Sara and her two friends bake gives 6 gumballs to Amber. a pan of 12 brownies. If the girls How many gumballs does Mark share the brownies equally, how have left? many will each girl have? 2. Lauren scored 6 points during 6. Erica ate 12 grapes. Riley ate the soccer game. Diana scored 3 more grapes than Erica. How twice as many points. How many grapes did both girls eat? many points did Diana score? 3. Kyle has 12 baseball cards. 7. Alice the cat is 12 inches long. Jason has 9 baseball cards. Turner the dog is 19 inches long. How many do they have How much longer is Turner? altogether? 4.
    [Show full text]
  • Sets and Boolean Algebra
    MATHEMATICAL THEORY FOR SOCIAL SCIENTISTS SETS AND SUBSETS Denitions (1) A set A in any collection of objects which have a common characteristic. If an object x has the characteristic, then we say that it is an element or a member of the set and we write x ∈ A.Ifxis not a member of the set A, then we write x/∈A. It may be possible to specify a set by writing down all of its elements. If x, y, z are the only elements of the set A, then the set can be written as (2) A = {x, y, z}. Similarly, if A is a nite set comprising n elements, then it can be denoted by (3) A = {x1,x2,...,xn}. The three dots, which stand for the phase “and so on”, are called an ellipsis. The subscripts which are applied to the x’s place them in a one-to-one cor- respondence with set of integers {1, 2,...,n} which constitutes the so-called index set. However, this indexing imposes no necessary order on the elements of the set; and its only pupose is to make it easier to reference the elements. Sometimes we write an expression in the form of (4) A = {x1,x2,...}. Here the implication is that the set A may have an innite number of elements; in which case a correspondence is indicated between the elements of the set and the set of natural numbers or positive integers which we shall denote by (5) Z = {1, 2,...}. (Here the letter Z, which denotes the set of natural numbers, derives from the German verb zahlen: to count) An alternative way of denoting a set is to specify the characteristic which is common to its elements.
    [Show full text]
  • Reihenalgebra: What Comes Beyond Exponentiation? M
    Reihenalgebra: What comes beyond exponentiation? M. M¨uller, [email protected] Abstract Addition, multiplication and exponentiation are classical oper- ations, successively defined by iteration. Continuing the iteration process, one gets an infinite hierarchy of higher-order operations, the first one sometimes called tetration a... aa a b = a b terms , ↑ o followed by pentation, hexation, etc. This paper gives a survey on some ideas, definitions and methods that the author has developed as a young pupil for the German Jugend forscht science fair. It is meant to be a collection of ideas, rather than a serious formal paper. In particular, a definition for negative integer exponents b is given for all higher-order operations, and a method is proposed that gives a very natural (but non-trivial) interpolation to real (and even complex) integers b for pentation and hexation and many other operations. It is an open question if this method can also be applied to tetration. 1 Introduction Multiplication of natural numbers is nothing but repeated addition, a + a + a + ... + a = a b . (1) · b terms | {z } Iterating multiplication, one gets another operation, namely exponentiation: b a a a ... a = a =: aˆb . (2) · · · · b terms | {z } Classically, this is it, and one stops here. But what if one continues the iteration process? One could define something like aˆaˆaˆ ... ˆa = a b . ↑ b terms | {z } But, wait a minute, in contrast to eq. (1) and (6), this definition will depend on the way we set brackets, i.e. on the order of exponentiation! Thus, we have to distinguish between the two canonical possibilities aaa..
    [Show full text]