Core Module 4: Sets, Functions, and Relations

To-Do Date: Feb 6 at 11:59pm

This is a two-week module with 3 class meetings. Exam 1 will happen in class on 2/14 (Wednesday of the second week). This module is not part of Exam 1. 

There are two parts of required readings for this module, both covered in this post, each corresponding to a prepare quiz. We expect these materials to roughly correspond to the first two class meetings of this module (i.e., 2/7 and 2/9).

The third class meeting, 2/16, is after the exam. There is no required reading for the 2/16 class meeting. This is because:

  • Making you read new material between 2/9 and the exam on 2/14 distracts you from preparing for the exam.
  • You deserve a break after taking the exam.

Therefore, for the 2/16 class meeting, just show up in class with a fresh mind and learn everything from scratch in the lecture. This is also how we will run all of the EMs, but the 2/16 class is NOT part of an EM; it is part of Core Module 4: Sets, Functions, and Relations.

  • Reading Part 1: AIDMA Chapter 4.1 – corresponding to CM4: Prepare Quiz I (Sets) and the 2/7 class.
    • In Example 4.10 the textbook says “If it seems strange to talk about whether or not two infinite sets have the same number of elements, don’t worry too much about it. We probably won’t bring it up again.” That might be true for the AIDMA textbook, but we do plan to learn more about infinite sets and their cardinalities in the 2/16 class (so you can safely ignore that topic before Exam 1).
  • Reading Part 2: AIDMA Chapter 4.3-4.4 – corresponding to CM4: Prepare Quiz II (Functions and Relations) and the 2/9 class. Some important notes about definition inconsistencies between AIDMA, MCS, and this imperfect world:
    • Definition of functions.
      • In AIDMA and MFADM, functions are defined first, and relations are defined as a separate concept. It’s (hopefully) clear that some relations are functions and some are not.
      • In MCS, relations are defined first, and functions are defined as functional relations. (You can read MCS Chapter 4 if you’re curious, but I don’t recommend.)
      • How MCS defines functions is not how the rest of the world defines it. For the rest of the world to call a relation a function, it must be both functional and total. (This includes AIDMA and MFADM, although they do not explicitly use the words functional and total in how they introduce functions.)
      • Bottom line: it is usual for a math concept to be not uniformly agreed upon. We should be thankful when it is and accept it (and stay vigilant) when it is not.
    • Definition of partial relations. 
      • In both AIDMA/MFADM, partial orders are defined as reflexiveanti-symmetric, and transitive.
      • In MCS, the same concept is explicitly called a weak partial order, which implies there is also a concept known as a strict partial order.
      • Elsewhere in the world, people sometimes define partial orders in other ways (e.g., just anti-symmetric and transitive).
    • We will sort these all out in class, so you do not need to worry too much about these discrepancies. 

To earn a satisfactory completion for CM4:

  • Get 80% or more questions correct in both CM4: Prepare Quiz I (Sets) and CM4: Prepare Quiz II (Functions and Relations)
  • Get a completion on recitation work by either attending or submitting on Gradescope
  • Get a satisfactory or above on the Gradescope assignment
  • Get all questions on the PrairieLearn homework (unlimited tries until LDoC)OR
  • Get an excellent on all Gradescope assignment questions
  • Get all questions on the PrairieLearn homework (unlimited tries until LDoC)

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