May I ask how to understand this result in Lesson 11 Binomial Identity?

In lesson 7, it says "Intersection is a higher priority operation than union."

Anyone can please explain why?

Hi Tianyi,

I can confirm Phil's answer.

First, it's important to understand precedence is a convention, not a theorem. We need precedence so that our expressions behave as intended. PEDMAS is a convention that is followed for common operations, but it does not covers all mathematical operations. Mathematics contains many rings outside of the set of reals with addition and multiplication. It is good practice to follow a similar precedence in all rings.

To help you see the similarities between intersection and multiplication, we can ask:

Consider the equality {} ∩ A = {1}. What is the set A? This is indeterminate, as this is equivalent to a division by 0.

Overall, precedence is a convention rooted in group theory. The precedence of set operations has to do with its definition in group theory and its relation to the PEDMAS convention.

Best, Marc Vicuna

I'm pleased to announce Introduction to Discrete Mathematics is now available on Wolfram U, including all course videos, exercises, quizzes and the final exam! Click Track My Progress to chart your certification progress. Go to the free course by visiting the course landing page and signing in using your Wolfram ID. Congratulations to Marc Vicuna and the Algorithms R&D team on this wonderful course, and kudos to the Wolfram U team for bringing it to fruition. Read Marc's blog post about the launch of the course: https://blog.wolfram.com/2023/11/29/dont-be-discreet-and-learn-discrete-mathematics-with-wolfram-language

A reminder to the Discrete Math Study Group participants that the deadline to pass the six quizzes for your course completion certificate is Friday, November 10. The exam can be taken anytime from the course framework. Refer to your Study Group emails for a link to the pre-release framework and to recording links from Study Group sessions. We heard a lot of positive feedback from Friday's review session. You might want to watch that recording, in particular.

Hello Study Group,

To confirm, the last question of lesson 22 lead to some confusion with reason, as the definition given for the vertex degree was wrong, or more precisely, only appliable to simple graphs. Therefore, the definition will soon be changed to "The number of incident edges to v is its degree.", since it's the number of edges that is important. This will be corrected.

Thank you for noticing and sorry for the confusion,

Marc

Hello Phil,

Let's go through the basics of the LinearRecurrence function.

Let's first consider the second input, which corresponds to the first terms of the sequences, a.k.a. base cases of the recurrence. Why specify that? Because all recurrences are based on n base cases, where n is the order of the recurrence.

Let's now consider the first input, the coefficients of the linear recurrence. This list is the coefficients of the terms in the recursive call, a.k.a. the recursive function (in f[n]= a f[n-1] + b f[n-2] + c f[n-3], the coefficients are {a, b, c}). Thus, the length of the list of coefficient specifies the order of the linear recurrence.

I'll now answer your question directly. Many basic sequences, such as when dealing with n-faced polyhedra, hypercubes, or many aggregation phenomena in data science may require special first terms, since the 0 or 1 case of applied phenomena often follow altogether different rules than the recurrent cases. If I have in a stack data structure the elements {5, 4, Q, 2, 1, 6}, and every time the stack is used, it read the head of the stack, and pushes double of the value into the stack, then after four operations I will have a stack of the sequence you made.

Another interesting case would be with multiagent system with limited knowledge, where the variable length of the list coefficients over the same list of base cases (environment, or prior knowledge) may express their degree of limited understanding of a given sequence.

Since the world is complicated and messy, especially in the initial cases, this feature is useful as it does not force your entire sequence to be dictated by the recurrence formula, exactly as an actual functional implementation would allow:

f[0]:=5;
f[1]:=4;
f[2]:=Q;
f[3]:=2;
f[4]:=1;
f[5]:=6;
f[n]:=2 f[n-1];

This is a valid implementation, and it is possible to represent it using LinearRecurrence, making it more programmer-friendly.

Best, Marc

Hi Tianyi,

This rule has many different proofs, which may depend on the equivalences you have access to in this context. I would generally use a proof by contradiction for both sides, as seen in this proof, where the perpendicular lines represent a contradiction:

This proof and other proofs are available here.

Best, Marc

I'm not seeing that, Ray. Today's (Monday, 10/23) link worked just fine for me.

At the beginning of the session @Marc Vicuna says that "the simplification of [propositional] expressions is not intuitive to a computer" -- implying that it is essentially impossible for the Wolfram Language (or any other computational language) to provide a natural intuitive way to simplify/process/visualize those propositions.

Mark, can you prove your claim? Can you prove that it's impossible to provide any sort of way to simplify/process propositions beyond what's available now? If not, what's the point of making such a claim?

The reason for my skepticism: I have seen the wondrous ways that the WL can provide simplifications in many domains. Those things were "impossible" -- until they were done in the Latest & Greatest version of the Wolfram Language. If there's some hard and fast reason why it can't be done here, it'd be educational to understand that reason why Discrete Mathematics is somehow different. TY.

Ray are you using Windows? Maybe the issue with the Lesson 1 notebook is happening because the filename has a question mark at the end. Most archivers have a problem with it as Windows does not allow filenames with a question mark.

7-Zip can handle files with invalid Windows filename characters and replace them with a "_" instead.

I assume Wolfram Research uses Macs where questions marks can be used in filenames.

Hello Ray,

Yes I see that file also, I'll remove it. It will be corrected.

Thank you, Marc

To answer a Q/A that appeared at the end of class today,

You can change the printing of any notebook using File->Print Settings->Print Setup or File->Print Settings->Print Setup->Printing Environment.

Personally, for my slides, I usually use a landscape print setup and an Elegant Printout printing environment, and then select the slide I want to print since there will be empty pages.

Feel free to experiment with the settings, and checking your result with File->Print Preview

Best, Marc

Hello Ray,

Indeed, it should be 46. Thank you for bringing attention to that.

Best, Marc

@Phil Earnhardt, you're correct, we mixed up our tagging system for Wolfram Daily Study Groups. The community tag has usually referenced the year, but we also use WSG code numbers (mostly internally) to track our Study Groups in the order they are scheduled and organized by Wolfram U staff. The Introduction to Discrete Mathematics is our 46th Wolfram Study Group, so that's where that tag came from. Thanks for being an active participant in this initiative.

Hello Ray,

It seems strange you don't have access to Lesson 1, I see it on the shared folder for the study group. In the mean time, here is lesson 1 is here in cloud and attached too.

Best, Marc

Attachments:

Lesson 1.nb

WRT resources in the course, one excellent resource I have found is Wolfram Mathworld: the web's most extensive mathematics resource. For definitions of terms, it's quite a winner. It's far superior to the Wikipedia: a small number of curators help insure high quality. I've stumbled across a fair number of wrong entries in the Wikipedia -- even in science articles -- but I've never seen anything but top-shelf data in Wolfram Mathworld. Note: all of the entries are Wolfram Language notebooks (!!!) available for download and manipulation.

Like many things associated with Stephen Wolfram or Wolfram Research, there's no good reason for Mathworld to exist. An individual created it in the 1990s, and Wolfram bought it in 1999. Entries continue to be added and enhanced. I first stumbled across the encyclopedia earlier this year when I as looking for a particular 3D Curve and found Viviani's Curve in the entries.

Looks like we start with Lesson 12 tomorrow. What about Lesson 11?

Think I heard somebody was looking for references. Here is a general document about Discrete Mathematics found at Yale and distributed under Creative Commons Attribution-ShareAlike 4.0 International license. The link: Notes on Discrete Mathematics

Hello!

Marc

Thank you, Lori.

This is noted. The poll questions are only used for the study group, so this will not appear on the course framework. However, my courses do tend to use a lot of images. It will be corrected. If you notice any similar problems in the lessons, exercises, or quizzes, feel free to tell us when we don't catch those mistakes ourselves.

Best, Marc

A real "Traveling Salesman" problem!

Recently, good friend Gil Hedley started his Nerve Tour, a series of presentations in 111 cities. Gil is conducting the tour over two years; he and his wife are traveling to all the North American cities with a camper. If you look on the tour page, you'll see the cities are laid out in a [rather boring] table; I thought it would be fun to map them out. I copied his data and used GeoListPlot to plot it out:

The tour length was 21,447 miles! All things equal, an optimized tour was 12,520 miles. But all things are never quite equal: Gil arranged to be in northern latitudes in warm months and southern cities in cooler months. I tried to draw the optimized map; I'm still wrestling with GeoListPlot. I'll update the forum when I have that map.

Hello Dave,

The Wolfram Language does not have symbolic set computations available. In fact, the Wolfram language intentionally does not have a proper theoretical set implementation, since it would require many implementation choices on definitions that may not fit the user. However, you can implement set manually easily with Sort and DeleteDuplicates.

In a sense, this is one major use of boolean expressions: you may simplify and analyze set expressions in logical form, then, when comes to computing actually sets, you may transfer your logical expression to a set expression. The equivalences between set theory and logic guarantee this. See this post for example.

Best, Marc

Hi Marc,

Thanks for creating this great (refresher) course.

Below I share a few minor issues I found:

Issues with lesson notebooks

• The notebooks are created with a higher version of Mathematica and it hangs the Kernel start-up on my machine. Workaround: start Mathematica first, then open the notebook.
• With the Mathematica Welcome Screen open, double-clicking a notebook may not open it. Workaround: use Open from the Welcome Screen menu; locate the file and open it.
• Scrolling a notebook, may hang the Kernel after pressing the Enable Dynamics button. Workaround: do not Enable Dynamics until you're done scrolling.

BigMarker recording at timecode 25:56

In the recoding of Monday's Lesson 2 Slide 5 Example (timecode 25:56)

Should this be: "I did not pass the exam?"

Given the logic, should this be "I did not study for the exam, nor did I pass it"?

Hello Wolfram community!

This course will start in 15 minutes! If you're interested, join us now with the link above!

Happy learning to all!

Marc Vicuna has done a great job in developing this thoroughly up-to-date introduction to Discrete Mathematics.

I strongly recommend the Study Group to everyone!

Hello Lori,

Thank you very much for your active participation and support for these online courses!