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Hi Jürgen,

Thank you for the great question, this is likely useful for many.

Let's start with or. Usually, this operator is the simplest to recognize, you'll find it when using "or" and it is sometimes implied when a choice is given. However, in day-to-day language, it is often confused with xor, the exclusive or. In our course, every time either is used, it is xor, anywhere else, or means or. In real situation, you may have to require your client to specify what is meant.

For the case of "and", while "and" is often used, it is also often implied. For example, in the line "this excellent white shredded French cheese", this cheese is excellent AND white AND shredded AND French. When searching for "and", it's best to think about what is said to be simultaneously true, and put in a conjunction all those statements. When and is not used, synonyms include but, however, along with, plus, also, as a consequence, as well as, furthermore, including, moreover, together with. Note two notable ways to get a negative "and", in the form "not p and not q": "nor" and "neither" are usually used to denote a negative and.

By far, the most difficult operator to recognize is the "implies" operator. There is p is a sufficient condition for q, p only if q, p implies q, q whenever p, q is a necessary condition for p (i.e., if not q, then not p), or ¬ q→¬ p, q is a consequence of p, q follows from p, q if p, if not q, then not p. See this post. Generally, if you see "if", "implies", you are in a conditional statement. When analyzing conditional, be also careful to determine which is the condition, and which is the implication.

Logic is everywhere, so people have all kinds of ways of expressing their reasoning. Sometimes, it just takes patience.

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.

Registered! The best Group Theory thing I ever figured out on my own: the late Skwish Toy inventor Tom Flemons pointed out to me that his manufacturing operation made the toy with a single elastic line that was segmented when the balls were glued to the end of the rods. That makes the lines of Skwish an Eulerian Cycle. In my University class, all of the groups used as examples for cycles, etc., were flat (and boring!) drawings on a page; I suggested to my prof to use the Skwish geometry on a problem set to mix things up a bit. The Manhattan Toy product is the best-selling tensegrity model on the planet, but most of the customers seem to just chew on it. :)

Hello Lori,

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

Hi Marc,

The following request is related to Week1_Polls.nb -> Lesson 5 -> Question 2.

Life is difficult but* fun.*

Yesterday, I have learned that the English word but means AND in the logical language. Do other keywords, like but, exist to translate English sentences into logical statements? Do you have a list of such keywords? Can you please give me a reference?

English is not my first language. Therefore, support is highly appreciated.

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

I'm curious why the community page for this study group has WSG23 in the name rather than WSG46.

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.

I'm a bit confused by what LinearRecurrence[] is doing. The one slide in Lesson 19 was a bit fast; I experimented with it a bit this morning and read the help file, and it now mostly makes sense.

One thing that still doesn't make sense is what LinearRecurrence does with the list of initial values. If I do

LinearRecurrence[{2}, {5, 4, Q, 2, 1, 6}, 10]

the response is

{5, 4, Q, 2, 1, 6, 12, 24, 48, 96}

I see in the documentation that only the last Length[ker] elements are used in the calculation -- the other elements are simply copied to the output. What is the physical (i.e., mathematical) significance of doing that copying operation on that subset of the input list? In other words, is there an example where someone would actually have a list of initial values that didn't matter? I have a hard time wrapping my brain around a list of inputs that are simply ignored.

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 Marc,

I have a problem reproducing the hypergraph in Lesson 24 notebook, section "Final States". If I run

ResourceFunction[
  "WolframModel"][{{x, y}, {y, z}} -> {{x, y}, {z, y}, {y, w}, {w, 
    x}}, {{0, 0}, {0, 0}}, 11, "FinalStatePlot"]

The result is this hypergraph

Where I used the same rule as the example in the section before:

rule = {{x, y}, {y, z}} -> {{x, y}, {z, y}, {y, w}, {w, x}};

I suspect another rule was used when the notebook was created; probably similar to the one in the example of the ResourceFunction["WolframModel"] documentation: https://resources.wolframcloud.com/FunctionRepository/resources/WolframModel/

ResourceFunction[
  "WolframModel"][{{x, y}, {x, z}} -> {{x, z}, {x, w}, {y, w}, {z, 
    w}}, {{0, 0}, {0, 0}}, 11, "FinalStatePlot"]

Which gives the plot shown in the lesson notebook:

Hello Dave,

The explanation actually even simpler! You're using the rule from slide 10, but the rule is actually found on slide 9. In other words, you could evaluate slide 10, then 9, then 11, and it would work. The rule is rule = {{x, y}, {x, z}} -> {{x, z}, {x, w}, {y, w}, {z, w}}; (slide 9)

and not rule = {{x, y}, {y, z}} -> {{x, y}, {z, y}, {y, w}, {w, x}}; (slide 10)

But this is fair, this order doesn't make much sense. I guess I'll just change the variable name of slide 10 to "rule2" or such. This will be corrected.

Thank you for noticing, 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.

Hi Davendra!

I've enjoyed three of your courses. Nice to 'see' you again :-D

Hello Dave,

For the versioning, we're working on changing it to the correct Mathematica version. For dynamics, there is not much we can do about it, unfortunately.

For the mistakes in the slides, that is correct, thank you, it will be corrected.

Best, Marc

Hello Jürgen,

I'd argue this is the right output. In mathematics, usually, we usually indeed write:

However, in code, the Wolfram Language uses a different but equivalent notation, where the variable is written as a subscript. This will still evaluate as you would expect. For example, Reduce[Exists[x, x == -x]] return True.

Thank you Marc,

I will explore the route to simplify and analyze set expressions in logical form and transfer them to sets later.

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,

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

Sorry, I was able to download the individual Lesson 1 file so I had access. I just thought you might want to correct the zip file before you go production.

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

Download files link in chat results in 404 error today. Old link previously saved works fine.

Cassidy reported the link has been fixed.

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.

@ray chandler Why should the tag be WSG46? I thought the tags for the WSG were always WSGYY where YY are the last 2 digits of the year when the WSG was held. This pattern has been followed for dozens of WSGs over the last 5 years.

What is the significance of the tag WSG46, Mark? What does that stand for?

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

Anyone can please explain why?

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

Hi Tianyi,

Honestly don't try to understand it, it seems to be a mistake. It took me a while to understand what was going on. I'm a mathematician in discrete mathematics, so I have limited knowledge of complex analysis. When I wrote that lesson, I was told this is the complex variant of the Vandermonde identity, related to the Chu-Vandermonde identity. After some digging, it seems this summation is wrong. I'll inform the Summation team to fix this.

Best, Marc Vicuna