Disjoint Sets

Randall Munroe (XKCD) recently posted a funny example of  two disjoint sets.   I've used one of his webcomics, with permission, in Chapter 1 of Applied Discrete Structures, so I probably won't use this but wanted to acknowledge it.


Link to this comic on XKCD

New in Version 3.5 of Applied Discrete Structures

The latest version of Applied Discrete Structures (3.5) has a few differences from the previous version.  Here is a list of the most significant changes:

In exercises:

• replaced 2.4 exercises 5 and 6 with lattice paths exercises
• added exercise 6 to section 4.1
• added bakery exercise at end of section 5.1
• added exercise #11 to 11.4
• exercise change 11.7 new #5, old #5 moved to #10
• changed exercise 4 of 13.4

Also, there is a new section (12.6)  on  linear equations mod 2.

The Ternary Number System

Every integer is the sum of distinct signed powers of three. That this is true is nice non-elementary, yet accessible, induction proof.  It's not an exercise in Applied Discrete Structures, but could be given as a challenge to your students.  The code implements the ideas of the proof, so it serves as a hint (or a solution?)

Several years ago, I had a WebMathematica page that computed the ternary representation of an integer.   That page is no longer in existence, and I discovered that it several broken links to it were on sequence pages of The On-Line Encyclopedia of Integer Sequences, such as sequence A072998.  As a replacement, I created a SageMath interact (similar to a Wolfram Demonstration).

The SageMath interact converts integers from 1 to 1000 to the ternary number system. It is contained within a Sage Cell, so anyone can view and tinker with the code. Any suggestions for improving it are welcome!

New version (3.5), Custom versions

Version 3.5 of Applied Discrete Structures is now in full distribution in all three formats, html, pdf and print. The full 16 chapters with solutions to half of the exercises runs over 600 pages.  For those who use the html format, the length doesn’t matter much; however, for the print and even the pdf format it may be desirable to have only a subset the chapters or sections.  

I’ve been distributing the book in two parts, corresponding to content that is normally covered in our two semester sequence at  UMass Lowell (Chapters 1-10 in Part 1- Fundamentals; Chapters 11-16 in Part II- Algebraic Structures).  Not many people buy these print versions, but I prefer the lighter books.   I don’t use them all that frequently since I tend to use html in class, but still use them to record typos (finally getting less frequent!) and making notes for improvements. 

One of the nice things about PreTeXt is that it’s quite easy to create custom versions.  If anyone would like some subset of the book in any format, let me know and I’ll create it. The only glitch I foresee is that if a reference is made to a non-included part of the text, a broken link will appear. I think that can be handled minimal extra work. 

Significant Additions in Version 3.5

Section 2.4: new exercises 5  and 6 on lattice paths

Section 5.1: added exercise 8 motivating the definition of matrix multiplication 

Section 11.4: added exercise 11 on inverting a linear function. 

New section: 12.6 Linear Equations over the Integers Mod 2

New Section: Linear Equations mod 2

The next version of Applied Discrete Structures will include a new section, 12.6, on systems of linear equations over \(\mathbb{Z}_2\).   We work with these systems in the coding theory section (15.5) but it was presumed that students could figure out how to solve these systems on the fly.  That is often the case, but some students had difficulties.

A pdf of the most recent draft of the new section is available at https://discretemath.org/Section_12_6_V2.pdf