HW01 - University of Illinois Fall 2009 ECE 313 Problem Set...

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University of Illinois Fall 2009 ECE 313: Problem Set 1: Solutions Sets, Events, Axioms of Probability and Their Consequences 1. [Subsets of a finite set] (a) The subsets of Ω = { ω 1 , ω 2 , ω 3 , ω 4 } are Subsets of size 0: Subsets of size 1: { ω 1 } , { ω 2 } , { ω 3 } , { ω 4 } Subsets of size 2: { ω 1 , ω 2 } , { ω 1 , ω 3 } , { ω 1 , ω 4 } , { ω 2 , ω 3 } , { ω 2 , ω 4 } , { ω 3 , ω 4 } Subsets of size 3: { ω 1 , ω 2 , ω 3 } , { ω 1 , ω 2 , ω 4 } , { ω 1 , ω 3 , ω 4 } , { ω 2 , ω 3 , ω 4 } Subsets of size 4: { ω 1 , ω 2 , ω 3 , ω 4 } = Ω There are 16 = 2 4 subsets of the set Ω of 4 elements. 15 = 2 4 - 1 of them are non-empty subsets. (b) OK, OK, geez, some people are never satisfied . . . (c) By checking our answers in parts (a) and (b), we see that there is: 1 subset of size 0 and 1 subset of size 4 - 0 = 4, 4 subsets of size 1 and 4 subsets of size 4 - 1 = 3, and lastly 6 subsets of size 2 with 4 - 2 = 2. For general n , whenever we choose a subset of size k , there is a unique subset of size n - k that is left out. That is, for every subset of size k , there is exactly one corresponding, complementary subset of size n - k , so the total number of subsets of size k is the same as the total number of subsets of size n - k . (d) i. The vector { 1 , 1 , . . . , 1 } corresponds to Ω. The vector { 0 , 0 , . . . , 0 } corresponds to . The n -bit vector with each bit flipped with respect to A corresponds to A c . ii. Since A B contains exactly those elements in either A or B , z i = x i y i . Similarly, since A B contains exactly those elements in both A and B , w i = x i y i . In purely arithmetical terms, z i = x i + y i - x i y i and w i = x i y i .

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