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Unformatted text preview: University of Illinois Spring 2010 ECE 313: Problem Set 1: Solutions Sets, Events, Axioms of Probability and Their Consequences 1. [Subsets of a finite set] (a) The subsets of = { 1 , 2 , 3 , 4 } are Subsets of size 0: Subsets of size 1: { 1 } , { 2 } , { 3 } , { 4 } Subsets of size 2: { 1 , 2 } , { 1 , 3 } , { 1 , 4 } , { 2 , 3 } , { 2 , 4 } , { 3 , 4 } Subsets of size 3: { 1 , 2 , 3 } , { 1 , 2 , 4 } , { 1 , 3 , 4 } , { 2 , 3 , 4 } Subsets of size 4: { 1 , 2 , 3 , 4 } = There are 16 = 2 4 subsets of the set of 4 elements. 15 = 2 4 1 of them are nonempty 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 . The vector { , ,..., } corresponds to . The nbit vector with each bit flipped with respect to A corresponds to A c ....
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