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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 1-4; Chapter 2, Sections 1-5 Powerpoint Lecture Slides, Sets 1-6 Noncredit Exercises: Chapter 1: Problems 1-5, 7, 9; Theoretical Exercises 4, 8, 13; Self-Test Problems 1-15. Chapter 2: Problems 3, 4, 9, 10, 11-14; Theoretical Exercises 1-3, 6, 7, 10, 11, 12, 16, 19, 20; Self-Test Problems 1-8 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 non-empty subsets? (b) If you listed only 14 or 15 subsets in part (a), please re-do 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 n-bit 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 one-to-one correspondence...
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- Spring '08
- Empty set, Natural number, total number, St. Ives