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MAT 3749 1.3 Part I Handout Just For Fun……ction? One-to-One Functions f : X Y is 1-1 (injective) if for each y Y , there is at most one x X such that f x y . Equivalent Criteria For x1 , x2 X , if f x1 f x2 then x1 x2 Last Edit: 4/30/2017 11:51 AM Example 1 Determine if the given function is injective. f : Z Z , f (n) 3n 1 Onto Functions f : X Y is onto (surjective) if the range of f is Y . Equivalent Criteria y Y , x X such that f ( x) y 2 Example 2 Determine if the given function is surjective. f : Z Z , f (n) 3n 1 Count Your Blessings… Bijections f : X Y is bijective if f is both 1-1 and onto. Inverse Functions If f : X Y is bijective, then its inverse function f 1 : Y X exists and is also bijective. 3 Equivalent Sets Two sets A and B are equivalent if there is a 1-1 function from A onto B . Example 3 The set of odd and even integers are equivalent. Analysis Proof Countable Sets A set A is countable if it is either finite or N is equivalent to A . Remark If an infinite set A is countable, then we can list its element as a sequence A a1 , a2 , a3 , . 4 Theorem The interval 0,1 is not countable. Analysis 5 Proof Suppose 0,1 is countable. Since 0,1 is not a finite set, the elements of 0,1 can be listed as a sequence an n 1 . For each of the real number an , we can represent it by its infinite decimal expansion. Thus, the sequence is a1 0.a11a12 a13 a 0.a a a 21 22 23 , * 2 a3 0.a31a32 a33 where each aij 0,1, 2, ,9 ij N . Construct a real number x 0.x1 x2 x3 such that n N , 5 ann 5 . xn 4 ann 5 Then, 1. x 0,1 . 2. n N , xn ann x an which is a contradiction to x 0,1 . Thus, 0,1 is not countable. Corollary The reals are equivalent to 0,1 , and are hence uncountable. HW 6 Theorem The set Q of rationals is countable. Proof Outline 1. List all the positive rational numbers as follows: 1 1 1 2 1 3 1 4 1 5 2 1 3 1 2 2 3 2 2 3 3 3 4 1 5 1 4 2 2 4 2. Form a sequence an n 1 by following the arrows and omitting any number that have already appeared. 3. The sequence 0, a1 , a1 , a2 , a2 , a3 , a3 , list all the rational numbers. 7 Classwork (Family of Sets) In this classwork, we are going to review the concept and notations of Family of Sets. Below, a, b and a, b represent open and closed intervals respectively. Recall Let A and B be two sets. A B x | x A or x B and A B x | x A and x B . 10 1. We know a n 1 n a1 a2 a10 . Let An be sets for n 1, 2,...,10 . What do you 10 think the notation An represents? n 1 2. Let An n, n 1 for nN . Find 1 1 3. Let Bn , for nN . Find n n 10 An . n 1 10 Bn . n 1 8 Variations 10 An for I 1, 2,...,10 An can be written as n 1 nI 4. Let An n, n 1 for nN and F 1, 2,8,9 . Find An . nF 1 1 5. Let Bn , for nN and F 1, 2,8,9 . Find n n Bn . nF 9 Here are the formal definitions for the union and intersection of families of sets. Definition Let F be a set and A n nF be a family of sets. Then, An x | x An for some n F and nF 6. Let Cy 2,sin y for y An x | x An for all n F . nF . (a) Make an educational guess on the explicit interval form of C y . No need to y actually prove your answer. You will do it in your HW. (b) Make an educational guess on the explicit interval form of C y . No need to y actually prove your answer. You will do it in your HW. 10