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Unformatted text preview: University of Illinois Spring 2010 ECE 313: Problem Set 1 Sets, Events, Axioms of Probability and Their Consequences Due: Wednesday September 2 at 4 p.m.. Reading: Ross Chapter 1, Sections 14; Chapter 2, Sections 15 Powerpoint Lecture Slides, Sets 16 Noncredit Exercises: Chapter 1: Problems 15, 7, 9; Theoretical Exercises 4, 8, 13; SelfTest Problems 115. Chapter 2: Problems 3, 4, 9, 10, 1114; Theoretical Exercises 13, 6, 7, 10, 11, 12, 16, 19, 20; SelfTest Problems 18 Yes, the reading and noncredit exercises are the same as in Problem Set 0. 1. [Subsets of a finite set] Let Ω denote a finite set containing the n elements ω 1 ,ω 2 ,...,ω n . The cardinality (more informally, the size ) of a subset A ⊂ Ω is the number of elements in A , and is denoted as  A  . (a) Let n = 4. List all the subsets of Ω in increasing order of size. How many subsets are there? How many of these subsets are nonempty subsets? (b) If you listed only 14 or 15 subsets in part (a), please redo part (a), and this time, include the empty set ∅ and/or Ω as subsets of Ω. (c) In your answer to part (a) or (b), verify that for each k , 0 ≤ k ≤ 4, the total number of subsets of size k is the same as the total number of subsets of size 4 k . Now explain why for n in general, the total number of subsets of size k is the same as the total number of subsets of size n k . (d) Each subset A corresponds to a nbit vector ( x 1 ,x 2 ,...,x n ) where x i = 1 if ω i ∈ A and x i = 0 if ω i / ∈ A . Writing A ↔ ( x 1 ,x 2 ,...,x n ) emphasizes the onetoone correspondence...
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This note was uploaded on 10/25/2010 for the course ECE 313 taught by Professor Milenkovic,o during the Spring '08 term at University of Illinois, Urbana Champaign.
 Spring '08
 Milenkovic,O

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