Set notation

Abstract: Some concepts and terminology of stratigraphy are expressed in terms of mathematical set notation. This formalized statement of traditional geologic concepts will facilitate computer manipulation of stratigraphic data. Set notation may be easily adapted to definition of rock units, stratigraphic order, stratigraphic correlation, time relationships, and fossil zonation.

Abstract: Introduction. It is well known that any set of 2 n - i elements of a solvable group of size n must contain a subset of size n with the property that the product of the dements of this subset arranged in the appropriate order yields the identity. In fact, if G is not cyclic, it is sufficient that the original set contain 2 n - 2 elements. The main result of this paper is that similar results hold for sets of size 2 n - r, where r is a fixed positive integer. The result says that for a fixed r, if n is sufficiently large and G is a non-cyclic solvable group, then any set of 2 n - r elements of G contains a subset of size n with the property that the product of the elements of this subset in the appropriate order is the identity. The term "set" in this paper actually means multi-set. That is, an element may occur more than once in a set. The cardinality of that set is number of elements in the set counting multiplicities. When we wish to consider an object which is a set in the classical sense, we shall use the term "ordinary set", All of the results in this paper concern sets of elements from a finite group. In some cases, the group is assumed to be abelian. In these cases, we shall use additive notation. In all other cases, multiplicative notation will be used. D e f i n i t i o n. Let G be a group and let S be a set of elements of G. A n-sum in S is an ordered subset of S of cardinality n. The result of that n-sum is the product of the elements in the n-sum in the specified order.

Abstract: Part 1 Sets: Set Notation, Cardinality and Membership. Sets With Changing Membership. Visualising Sets - Venn diagrams. Union, Intersection, and Set Difference. Predicate Definition of a Set. Sets Used To Describe Types. Pattern Definition of a Set. Combining Predicate and Pattern Definitions. Exercises. Answers. Part 2 Propositions and Further Sets: Propositions. Combining Propositions. Truth Tables. Priorities of Logical Connectives. Set Equality and Subsets. Power Set. Cartesian Product. Types Extended. Two More Local Connectives - Implication and Equivalence. Some Useful Laws. Rearranging Propositions. Exercises. Answers. Part 3 SCHEMAS: Formal Specifications. The Tape Collection. Types. Axiomatic Descriptions. Schemas. The Hotel Room Problem. Exercises. Answers. Part 4 Predicates: Predicates. For All Members Of A Set - Universal Quantification. There Exists - Existential Quantification. There Exists Exacty One. Propositions and Predicates in Use. Exercises. Answers. Part 5 Relations: Relations. Relations and the Cartesian Product. More Complex Relations. The Domain and Range of Relations. Domain Restriction. Domain Anti-restriction (Subtraction). Range Restriction. Range Anti-restriction. Relational Composition. Inverse of a Relation. Exercises. Answers. Part 6 Functions: Functions. Partial and Total Functions. Injective Functions. Surjective Functions. Bijective Functions. Function Inverse. Function Overwrite. Functions in Schemas. Exercises. Answers. Part 7 Sequences: Sequence Basics. Functions on Sequences. QUEUE Case Study. Exercises. Answers. Part 8 Specifying Functions: Format. Types of Specifications. Generic Functions. Example Function Specifications. Exercises. Answers. Part 9 Modelling: Models. Paper Model. Z Model. Exercises. Answers. Part 10 Case Study: The Case. State Specification. Operations. Exercises. Answers. Part 11 Schema Operators: Inclusion. Disjunction. Conjunction. Decoration Delta. xi E. Renaming. Hiding. Exercises. Answers. Part 12 Pre-condition analysis: Need for Pre-condition Analysis. Informally Determining the Pre-conditions of a Schema. Formally Determining the Pre-conditions of a Schema. Exercises. Answers. Part 13 Implementation: Stages of Implementation. Implementing Types. Refinement. Using Abstract Data Types. Exercises. Answers. Part 14 Epilogue: Multichoice Multichoice. Answers.

Abstract: Part A: The Foundations of Quantitative Methods Ch 1 Quantitative Methods and what this book is all about 11 What is Quantitative Methods? 12 How to use this book 13 Using calculators, computers and spreadsheets 14 Chapter review Ch 2 Quantities and Elementary Arithmetic 21 Number systems 22 Truncating, rounding and precision 23 Arithmetic operations and notation 24 Powers, roots and logarithms 25 Operator precedence 26 Fractions, ratios and percentages 27 Units of measurement 28 Chapter review Ch 3 Computing, Notational Systems and Algebra 31 Algorithms, Variables and Algebra 32 Simultaneous and systems of equations 33 Scalars, Vectors, Matrices and Arrays 34 Subscripted Variables and Sigma Notation 35 Chapter review Ch 4 Matrix algebra 41 Matrix addition and subtraction 42 Matrix multiplication 43 Special matrices 44 Using matrices to solve systems of equations 45 Finding the inverse and the determinant of a matrix 46 Chapter review Ch5 Sets and counting 51 Set Notation, Set Properties and Venn Diagrams 52 Counting, factorials, permutations and combinations 53 Pascal's triangle and binomial coefficients 54 Chapter review Ch6 Elementary statistics 61 The meaning and nature of Statistics 62 Sampling and data collection 63 Presenting and describing small data sets 64 Presenting and describing large data sets 65 Measures of location 66 Measures of variation 67 Index numbers 68 Chapter review Part A Closing comments Mini cases: Tops 'n' Bottoms, CompRus Ltd, JoowelsStore Part B: Models and Analysis for Business and Management Ch 7 Probability and Statistical Models 71 Probability 72 The Binomial distribution 73 The Poisson distribution 74 The Exponential distribution 75 The Normal distribution 76 Chapter review Ch 8 Introduction to inferential statistics 81 Estimating and Confidence intervals 82 An introduction to Hypothesis testing 83 Non-parametric tests 84 Chapter review Ch 9 Modelling simple relationships 91 Relationships, functions and equations 92 Linear functions 93 Least-squares linear regression 94 Appropriateness and correlation 95 Chapter review Ch 10 Non-linear and multivariate functions 101 Quadratic functions 102 Polynomial functions 103 Hyperbolic functions 104 Exponential functions 105 Multivariate functions 106 Chapter review Ch11 Analysing models 111 Graphical and numerical analysis 112 Marginal analysis and differentiation 113 Elasticity 114 Optimisation 115 Integration 116 Partial differentiation and applications 117 Chapter review Ch12 Time series analysis 121 Classical time series analysis 122 Methods for Classical time series analysis and forecasting 123 Exponential smoothing 124 Chapter review Ch 13 Models for Finance and Accounting 131 Simple interest and compound growth 132 Multiple compounding, APR and continuous compounding 133 Discounting and other related applications of compound growth 134 Savings, Endowments and Sinking Funds 135 Loans, Mortgages and Annuities 136 Chapter review Part B Closing comments Mini cases: NHTBeds, Firmecon, Mr and Mrs Lean Part C: Modelling and Analysing Decisions Ch14 An introduction to decision analysis 141 Decisions under certainty, uncertainty and risk 142 Decisions under uncertainty 143 Decisions under risk 144 Chapter review Ch15 Decisions in finance 151 Discounted Cash Flow and Net Present Value 152 Internal Rate of Return 153 Payback Period 154 Chapter review Ch 16 An Introduction to Mathematical Programming 161 An introduction to Linear Programming 162 Solving Linear Programs graphically 163 An introduction to the Simplex Method 164 An introduction to non-linear programming 165 Chapter review Part C: Closing Comments Mini cases: Makit Ltd, Pandbe Co, PrintRus Further Reading Tables Index