Exam1Spring08 - Math 213 Exam 1(Solutions Prof I.Kapovich...

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

View Full Document Right Arrow Icon
Math 213 Exam 1 (Solutions) Prof. I.Kapovich February 22, 2008 Problem 1 [20 points] For each of the following statements indicate whether it is true or false. You do not have to explain your answers. (1) For the set S = {∅ , {∅} , {{∅}}} we have | S | = 1. (2) If | A | = m and | B | = n , then the number of all functions from A to B is m n . (3) For every n r 1 we have C ( n, r ) P ( n, r ). (4) Whenever E, F are events such that p ( E ) = 0, then E and F are independent. (5) For every n 1 the number n i =0 ( n i ) is even. Solution. (1) FALSE. In fact, we have | S | = 3. (2) FALSE. In fact, he number of all functions from A to B is n m . (3) TRUE. Indeed, C ( n, r ) = P ( n, r ) /r ! and so C ( n, r ) P ( n, r ). (4) TRUE. Indeed, in this case p ( E ) = 0 and 0 p ( E F ) p ( E ) = 0, so that p ( E F ) = 0. Thus 0 = 0 · p ( F ), that is p ( E F ) = p ( E ) p ( F ). (5) TRUE. Indeed, n i =0 ( n i ) = 2 n is even. Problem 2 [20 points] Prove that for every integer n 1 we have: 1 2 - 2 2 + 3 2 - · · · + ( - 1) n - 1 n 2 = ( - 1) n - 1 n ( n + 1) 2 . Give all the details of your work.
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