34 is a rational number 2 is not a rational number

Info icon This preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: set of positive even integers E is a countable set. Solution: let f (x) = 2x. Z+: 1 2 3 4 5 6 … 2 4 6 8 10 12 … Then f is a bijection from Z+ to E since f is both one- to- one and onto. Posi$ve Ra$onal Numbers are Countable —༉  Reminder: a rational number can be expressed as the ratio of two integers p and q such that q ≠ 0. —༉  3/4 is a rational number. —༉  √2 is not a rational number. Example 2: show that the positive rational numbers are countable. Solution: the positive rational numbers are countable since they can be arranged in a sequence: r1 , r2 , r3 , … The next slide shows how this is done. → Posi$ve Ra$onal Numbers are Countable First row q = 1. Second row q = 2. etc. Constructing the List First list p/q with p + q = 2. Second list p/q with p + q = 3 Third list p/q with p + q = 4, and so on. Matrices —༉  Matrices are useful discrete structures that can be used in many ways. For example, they are used to: —༉  Describe certain types of functions known as linear transformations , e.g., they are used in computer graphics. —༉  Express which vertices of a graph are connected by edges. —༉  Later, we will see...
View Full 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