Saturday, August 12, 2023

New: Applied Discrete Structures is on Runestone Acadamy

Applied Discrete Structures is now available on
Runestone Academy. To access it, go to the Runestone Library Mathematics list

 At this point the book doesn’t have reading questions so the fully capabilities of Runestone are  not being fully utilized.  By 2024, I’m hoping to have added a pool of reading questions for most sections. This will allow students to test their reading comprehension in a low-stakes environment. Instructors can incentivize students’ reading by grading completion of reading assignments. Currently, a record is only kept of which sections of the book have been opened by each student. 

Applied Discrete Structures is still available in other form factors at

Friday, May 26, 2023

Version 3.10 of Applied Discrete Structures - May 2023

 The new version of Applied Discrete Structures has been released in all but the print version. The two most significant changes in this version are 

  • Equivalence classes are now defined within the text (Section 6.3) as opposed to being introduced in an exercise.
  • A glossary. Many of the words in this glossary are not formally defined in the book either because they are viewed as prerequisites to a course in discrete mathematics or are terms in computer science that some students may be unfamiliar with.
The usual minor typos have been corrected and a few exercises have been added or revised. 

The html and pdf versions are available now at  

The print version will be available later this summer. It’s available roughly at cost, but prices have been inching up in recent years.  Last year the full version was $43+shipping.  Seems high to me, but still not as bad as many of the prices for books offered by publishers. As expected, sales have also dropped. No big deal - the print form factor exists mostly to qualify the book for a few OER listings that require it.

Friday, October 14, 2022

A new matrix multiplication record

In Chapter 5, we mentioned Strassen’s algorithm for matrix multiplication. Not many improvements have been made since 1969 when Strassen discovered how to multiply a \pair of 4 by 4 matrices with 49 multiplications - a reduction of 15 multiplications from the  basic definition. A further reduction by one multiplication was announced by two Austrian researchers at Johannes Kepler University Linz in October 2022. 

Monday, September 26, 2022

Does learning about quantifiers help students understand limits?

 A recent thread on the MAA member web site discussed how limits should be taught in Calculus I/II. One comment was that students who take a discrete math course, where quantifiers are discussed, might better understand the definition of a limit.  What follows is a possible example that could be added to our section on quantifiers. Background:  I taught a calculus workshop for mostly middle school teachers several years ago and I recall the most spirited discussion being around the idea that $0.999…  = 1$.

Example: What does it mean that 0.999… = 1?  The ellipsis (…) implies that there are an infinite number of 9’s on the left of the equals sign.  After many years of struggling with what this means, mathematicians have come up with a universally accepted interpretation involving quantifiers.  It is that

$$(\forall \epsilon)_{\mathbb{R}^+} ((\exists N)_{\mathbb{P}})(n\geq N \Rightarrow  |1- 0.\underbrace{99..9}_{n\,9’s}| \lt \epsilon))$$

In calculus, the symbol $\epsilon$ is usually reserved for small positive real numbers. Let’s pick a value for $\epsilon$ and peel the universal quantifier off the statement above.  Let’s try  $\epsilon$ equal to $\frac{1}{2^{10}}=\frac{1}{1024}$.  In addition we note that $0.\underbrace{99..9}_{n\,9’s}=1-\frac{1}{10^n}$.  With our choice of $\epsilon$ we get 

$$ (\exists N)_{\mathbb{P}}(n\geq N \Rightarrow  |1- 0.\underbrace{99..9}_{n\,9’s}| \lt \frac{1}{1024}) $$

$$(\exists N)_{\mathbb{P}}(n\geq N \Rightarrow \frac{1}{10^n} \lt \frac{1}{1024}) $$

This last statement is true - one value of  $N$ that would work is $11$. You just have to convince yourself that any positive value of $\epsilon$, no matter how small, will produce a true statement.  If you see that, you’ve convinced yourself that $0.999…  = 1$!

Thursday, August 5, 2021

UMass Lowell Faculty Word Search

 We have another month before the semester gets started, so here is a word search puzzle you can do while you relax in your back yard or the beach. Most of the names are the last names of UMass Lowell Math Faculty and Staff. A few individuals had names too long for the puzzle app to handle, so their first names are used instead. Also, a few of the names are spelled right to left or bottom to top.