---
tags: mth350, dailyprep
---

# Daily Prep 5A -- MTH 350-01

## Overview

With Module 5, we begin a departure from the concrete world of integers into a more abstract point of view. A fundamental tool for us in that world is the **equivalence relation** (which you first met in MTH 210) and **equivalence classes**. These tools allow us to take a set of objects and declare when two of them are "basically alike" in a way of our choosing. We'll review the idea of an equivalence relation and prove some important fundamental results about equivalence classes, then apply the concepts to integer congruence.

## Learning objectives

**Basic Learning Objectives:** *Before* our class meeting, use the Resources listed below to learn all of the following. You should be reasonably fluent with all of these tasks prior to our meeting; we will field questions on these, but they will not be retaught.

+ Define the *congruence class* of an integer modulo n; use the definition to write out the congruence class of a specific integer given a specific value of n as a set in roster notation.
+ Define the concept of an *equivalence relation* on a set; decide if a given relation on a set satisfies the definition or not.
+ Define the concept of the *equivalence class* of an element of a set with respect to a given equivalence relation; use the definition to write out the congruence class of a specific object given a specific equivalence relation.
+ State the definition of the *integers modulo n* and identify this set by its symbol.

**Advanced Learning Objectives:** *During and after* our class meeting, we will work on learning the following. Fluency with these is not required prior to class.

+ State and apply the properties of congruence classes listed on page 46.
+ Perform addition and multiplication of congruence classes within $\mathbb{Z}_n$$\mathbb{Z}_n$ and make addition and multiplication tables for specific instances of $\mathbb{Z}_n$.
+ Explain line-by-line the proof that addition in $\mathbb{Z}_n$ is associative.
+ Prove or disprove that the analogues of Axioms of Arithmetic hold for $\mathbb{Z}_n$.

## Resources for learning

**I recommend watching the videos first this time, then scanning through the reading.**

**Video:**

- Relations (6:53) https://www.youtube.com/watch?v=qnjxdlpWMLA&list=PL2419488168AE7001&index=97
- Properties of relations (9:55) https://www.youtube.com/watch?v=rCuPn0GZ7J8&list=PL2419488168AE7001&index=99
- Equivalence relations (14:03) https://www.youtube.com/watch?v=JFXgXYCzXB4&list=PL2419488168AE7001&index=100
- Equivalence classes (5:54) https://www.youtube.com/watch?v=-C6Rnk0W2lE&list=PL2419488168AE7001&index=101
- The integers mod n (7.4.1) https://www.youtube.com/watch?v=Tl9Xxdoeg7I&list=PL2419488168AE7001&index=104

**Reading:** Read through Investigation 5 (pp. 59--69) in the textbook.

## Exercises

The exercises for this Daily Prep are found on the Google Form:

https://docs.google.com/forms/d/e/1FAIpQLSdqQcfqkIUYJg-jjZGl9aIjoSEyvV2lGV-_buRxNQj0eKweZA/viewform

## Submission and grading

To submit your work, simply submit the Google Form. You will receive a receipt via email to confirm your submission. (Look in your spam folders if you do not see the receipt.)

A **Pass** mark is given if the Daily Prep is turned in before its deadline and if each item on the Daily Prep has a response that represents a good faith effort to be right. **Mistakes are not penalized**. A **No Pass** is given if an item is left blank (even accidentally), has an answer but it shows insufficient effort (including responses like "I don't know"), or if the Daily Prep is late.