# Introduction to Set Theory,Revised and Expanded by Wendy Willard

By Wendy Willard

Completely revised, up-to-date, accelerated, and reorganized to function a major textual content for arithmetic classes. DLC: Set conception.

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E.. by starting with 0 and repeatedly applying the successor operation: 0. 0 + 1 = 1,1+1=2,2+1=3,3+1=4,4+1=5 , . . etc. 2 Definition A set I is caIled inductive if (a) 0 E I. (b) If n E I , then ( n+ 1) E I. An inductive set contains 0 and, with each element, also its successor. cording t o (c), an inductive set should contain all natural numbers. 'fhe precis? , it is the smallest i~~clucstivt. set. This leads t o the following definition. 1. 3 Definition The set of all natural numbers is the set N = {X IX E I for every inductive set The elements of N are called natural numbers.

8 Definition Let S be a partition of A. The relation Es in A is defined by Es = {(a, b) E A X A I there is C E S such that a E C and b E C). Objects a and b are related by Es if and only if they belong to the same set from the partition S. 9 Theorem Let S be a partition of A; then Es is an equivalence on A. Pmof. (a) Reflezivity. Let a E A; since A = US, there is C E S for which a E C , so ( a ,a ) E Es. (b) Symmetry. Assume aESb;then there is C E S for which a E C and b E C. Then, of course, b E C and a E C, so bEsa.

This is not an accident. 5 Theorem Let f and y be functions. Then y o f is a functior~. g o j' i s defined at X if and only if f is defined at X and y is defined at f ( x ), 2 . , dorn(g o f ) = dom f 17f-'[domg]. T E ciom(g o f ). 26 CHAPTER 2. RELATIONS, FUNCTIONS, AND ORDERINGS Example Let f = (1/x2 ] X # 0);find f a ' . As f = { ( x , l / x " ) x # 0}, we get f - l = {(1/x2,X) 1 X # 0). f - l is not a function since ( 1 . -1) E f - l . ( 1 , l ) E f - l . Therefore, f is not one-to-one; ( 1 , l ) E f .

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