Berkeley MATH 113 - Math Language (12 pages)

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Math Language



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Math Language

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Pages:
12
School:
University of California, Berkeley
Course:
Math 113 - Introduction to Abstract Algebra
Introduction to Abstract Algebra Documents
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George M Bergman Math H110 Fall 2008 Supplementary material Some notes on sets logic and mathematical language These are generic notes for use in Math 110 113 104 or 185 These pages do not develop in detail the definitions and concepts to be mentioned That is done to various degrees in Math 55 Math 74 Math 125 and Math 135 I hope you will nevertheless find them useful and thought provoking I recommend working the exercises for practice but don t hand them in unless they are listed in a homework assignment for the course 1 Set theoretic symbols Symbol Meaning usage examples discussion The empty set Here denotes the set of all natural numbers i e 0 1 2 3 while from Zahl German for number denotes the set of all integers 3 2 1 0 1 2 3 Many older authors started the natural numbers with 1 but it is preferable to start with 0 since natural numbers are used to count the elements of finite sets and is a finite set Of these for quotient denotes the set of all rational numbers i e fractions that can be written with integer numerator and denominator denotes the set of real numbers and the set of complex numbers The five sets just named used to be and often still are denoted by bold face letters N Z Q R and C The forms arose as quick ways to write these boldface letters on the blackboard Since it is convenient to have distinctive symbols for these important sets printed forms imitating the blackboard bold symbols were then designed and are now frequently used as shown Is a member of E g 3 The set of all This is often used together with or different authors prefer the one or the other which stand for such that For instance the set of positive integers can be written 1 2 3 or n n 0 or n n n 0 The set of all square integers can be written 0 1 4 9 n 2 or n 2 n Note also that n 2 n m 2 m m 2 2m 1 m Why Is a subset of E g 1 n 2 n n n 0 or In fact Is a proper subset of that is a subset that is not the whole set For instance all the formulas used above to illustrate remain true with in place of except for Since a proper subset is in particular a subset one may use even when is true and one generally does so unless one wants to emphasize that a subset is proper But beware some authors especially in Eastern Europe use for is a subset of etc Obvious variants of the above symbols A B means B A x X means x is not a member of X A B means A is not a subset of B etc Warning The phrases A lies in X and A is contained in X can mean either A X or A X The former phrase more often means A X and the latter A X but this is no guarantee So in your writing if there is danger of ambiguity either use the mathematical symbols or use the unambiguous phrases is a member of and is a subset of Intersection A B x both x A and x B are true For instance x x 9 x x 12 10 11 12 2 Intersection of an indexed family of sets For instance if A 0 A1 are sets then n 0 1 An also written n An n 0 An and A 0 A1 An means x x is a member of all of A 0 A1 etc In an intersection i I Ai I does not have to be a set of integers it can be any set such that Ai is defined for each i I When a set I is used in this way to index i e list other entities it is called an index set Union A B x x A or x B For instance x x 0 x x 12 Note that if A B then A B B and A B A Union of an indexed family of sets Thus n 0 1 An n An n 0 An A 0 A1 An Example n i i n Often when the intent is clear from context the above notations are abbreviated For instance if we know that we have one set Ai for each i in a certain index set I then instead of i I Ai we may write I Ai or i Ai or just Ai Likewise if we have a set Xn for each positive integer n the intersection n 1 Xn may be written n Xn or simply Xn c Complement of c A means x x A But x x A makes no sense if we don t say what x s are allowed So the notation c A is used only when discussing subsets of a fixed set For instance if we are discussing subsets of the integers then c even integers odd integers while if we are considering subsets of then c even integers denotes the set consisting of all odd integers and non integer rationals To be more precise we can use the next symbols or f X Y even integers odd integers Note that A B or A B means x A x B E g A B is defined even if B is not a subset of A For instance negative real numbers This indicates that f is a function also called a map or mapping from the set X to the set Y In reading the symbol out loud one can use words such as the map f from X to Y or f sending X to Y Such an f is said to be one to one or injective if for every two distinct elements x1 x 2 X the elements f x1 and f x 2 of Y are also distinct For instance the operation of cubing an integer is a one to one function but the squaring map is not one to one because n 2 n 2 The function f X Y is said to be onto Y if every element of Y equals f x for some x X For instance the squaring and cubing maps are not onto since not all integers are squares or cubes On the other hand the cubing map is both one to one and onto Given f X Y the set X is called the domain of f What about the set at the other end of the arrow A complication is that if f is not onto Y then Y and f x x X are different sets Traditionally these were called the range and the image of f respectively but the usage was not firm range was often used as a synonym for image Hence nowadays the unambiguous term codomain has been introduced to describe Y A function is called onto if it is onto its codomain a synonymous term is …


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