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...

• Tarscor
• 2

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!

• Spring '08

What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern