QuestionQuestion

Transcribed TextTranscribed Text

1 Functions (15 P.) Prove or disprove the following statements. a) (4) Given the family of sets N%a = {x 2 N| x mod a = 0} with a 2 N+. There exists a bijection f : N%x ! N%y for any pair x, y 2 N+. The set N%x is the set of all (positive) multiples of x. So the set can also be written as {x, 2x, 3x, 4x, 5x, . . . }. The same goes for N%y = {y, 2y, 3y, 4y, 5y,... }. Now, you only need to define an identity function, which maps x to y, 2x to 2y and so on. I’ll leave it to you as a task to find this identity function and prove its bijectivity. b) (4) The function fn : Nn ! Nn x 7! (xn + 1) mod n is bijective with Nn = {x 2 N| x  n} for every n 2 N+. The set Nn are just all natural numbers less or equal to n. So, N5 = {0, 1, 2, 3, 4, 5}. The function fn has the set Nn as domain and codomain. Since the set Nn is finite for any n 2 N+, the function is either bijective or neither injective nor surjective (but never just one of the two). As a last hint: The function is not bijective. I’ll leave it up to you, to find a respective counterexample. c) (7) There exists a bijection f : N ! N ⇥ N You can imagine N as dots on the number-line and N ⇥ N as a grid of dots in a 2D plane (only covering the upper right). This grid is also indicated in Figure 1. Figure 1: Illustrative idea of how N ⇥ N looks like From here on, you just have to find a plausible way to order these points, like “this is the first one, this is the second one, that’s the third one” and so on. If you can find such a mapping, you also have to show it’s bijective, but that’s again a task left for you. 1

Solution PreviewSolution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

    By purchasing this solution you'll be able to access the following files:
    Solution.pdf.

    $13.00
    for this solution

    or FREE if you
    register a new account!

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Discrete Math Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Decision:
    Upload a file
    Continue without uploading

    SUBMIT YOUR HOMEWORK
    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats