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

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 .
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern