---
tags: mth350, dailyprep
---

# Daily Prep 7B -- MTH 350-01

## Overview

Module 7B continues the exploration from Module 7A, where we are looking at various number systems to see how many of the Axioms of Arithmetic they satisfy. In this module, we're going to exit the world of numbers altogether and look at a "number" system composed entirely of abstract sets, with the operations of addition and multiplication being replaced with symmetric difference and intersection, respectively. How will this "number" system work with these two operations? What does arithmetic look like when you're no longer using numbers?

## 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.

- (Review) Given a finite set, list all the elements of its *power set*.
- (Review) Find the intersection and the *symmetric difference* of two (finite) sets.
- Write out the elements of the set $\cal{P}_n$ where $n$ is a natural number.

**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.

- Given a finite set (of a reasonably small cardinality), write out its power set and then construct operation tables for symmetric differences and intersections.
- Determine which Axioms of Arithmetic are satisfied for $\cal{P}_n$ using symmetric difference as "addition" and intersection as "multiplication". In particular, identify the additive and multiplicative identities of these two operations.
- Identify the units and zero divisors of $\cal{P}_n$ for different values of $n$ and make conjectures about these objects in general.
- Compare the operation tables for $\cal{P}_n$ with other finite number systems and note the similarities and differences.

## Resources for learning

**Reading:** Read pages 70-71 in the textbook, which is just the section called "Collections of Sets". This section defines all the terms that you need to know and gives a little example.

**Video:** No video this time.

## Exercises

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

https://docs.google.com/forms/d/e/1FAIpQLSfuvCZWm7Zl5vY5OFRZErXWwo6-AhJ8Y4LPO7w8udVCw-vmCg/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.