dis6b-sol.pdf - CS 70 Fall 2019 Discrete Mathematics and Probability Theory Alistair Sinclair and Yun S Song Quiz 6 1[True or False(a The set of all

dis6b-sol.pdf - CS 70 Fall 2019 Discrete Mathematics and...

This preview shows page 1 - 2 out of 2 pages.

CS 70 Discrete Mathematics and Probability Theory Fall 2019 Alistair Sinclair and Yun S. Song Quiz 6 1. [True or False] (a) The set of all irrational numbers R \ Q (i.e. real numbers that are not rational) is uncountable. (b) The set of integers x that solve the equation 3 x 2 ( mod 10 ) is countably infinite. (c) The set of real solutions for the equation x + y = 1 is countable. For any two functions f : Y Z and g : X Y , let their composition f g : X Z be given by f g = f ( g ( x )) for all x X . Determine if the following statements are true or false. (d) f and g are injective (one-to-one) = f g is injective (one-to-one). (e) f is surjective (onto) = f g is surjective (onto). Solution: (a) True. Proof by contradiction. Suppose the set of irrationals is countable. From Lecture note 10 we know that the set Q is countable. Since union of two countable sets is countable, this would imply that the set R is countable. But again from Lecture note 10 we know that this is not true. Contradiction!
Image of page 1
Image of page 2

You've reached the end of your free preview.

Want to read both pages?

  • Spring '08
  • PAPADIMITROU

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

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask You can ask (will expire )
Answers in as fast as 15 minutes
A+ icon
Ask Expert Tutors