PreTeXt Showcase

Many mathematics papers and textbook chapters are organized in the traditional definition-lemma-proof-theorem-proof format. Depending on how friendly the paper is, or who the intended audience is, there might be quite a bit of exposition surrounding these elements. This subsection is intended to be an illustration of this structure.

We begin with some basic definitions. Depending on the narrative style of your text, you might want to define terms in the middle of paragraphs. We might call this an inline definition, which can be accomplished using the <term> tag.

More formal texts will likely want numbered definitions.

One advantage to having numbered definitions (see Definition 2.1) is that you can refer to them, just like we just did. We can also refer to titled definitions (see Definition 2.2).

At this point it might be useful to give a few examples of the terms defined by your definitions. Of course here the definitions are examples of definitions, so instead will illustrate our point with an example of an example.

While on this topic, it is worth mentioning that many authors choose to include examples of problems that can be solved. This suggests that part of an example is the solution. For example:

Returning to our main goal of establishing new mathematical theory, we might next state an axiom or two (especially if we are named Euclid). In PreTeXt the environments for axioms are not “definition-like” (the only definition-like environment is a definition). There are quite a few “axiom-like” environments.

Once you have defined your terms and stated your axioms, you will likely want to prove something. Perhaps you will start with a lemma or two, use them to prove a theorem, followed by a few corollaries. If you are not as confident in the importance of your results, you might call them propositions or just facts.

The lemma above did not have a proof, since it’s proof is obvious.

Using the two lemmas above, we arrive at the following:

There is no reason you need to prove a theorem right away. You can always have some text, like this paragraph, before the proof. You could even have an example of how a theorem could be applied before pulling the reader back in for the eagerly awaited proof.

Note that if you don’t give a proof a title, it will get titled “proof” automatically. It makes sense to put a custom title in this case since you are referring to a proof of the a theorem that was stated without proof earlier.

Lemmas, theorems, and corollaries naturally go together. There are other theorem-like environments that you might view as parallel to these. Here is an example.

If students are reading your textbook, it might be time to collect the main points that you have made for your students and put them in a box. You could also put your result in the box and let your student roam free. The following is an <assemblage>.

Your paper will likely have lots of remarks about its content as the connecting text between theorems, as this section exemplifies. However, it might also be useful to highlight important remarks, or to set off remarks that do not otherwise fit into the main flow of the text.

If your remark is far removed from your main content, you might make it as an aside.

No self-respecting paper on self-reference would be complete without some open questions. You might make some conjectures, which are axiom-like environments in PreTeXt.

You might also present a question for your readers. The <question> and <problem> environments might be appropriate, although note that these are example-like, as they can receive hints, answers, and solutions.

As a final example, we present a renamed environment. In PreTeXt, one of the theorem-like environments is fact. But perhaps you would rather use “Self Evident Fact” as your environment name. The following is coded as <fact>, but using <rename> in <docinfo> we can get the following

In addition to <example>, PreTeXt provides two more “example-like” environments: question and problem.

Another way you might structure a chapter is to give students more opportunities to actively engage in the material. You might give a long list of definitions, and then a list of theorems to prove. That would look a lot like the previous section (perhaps with fewer paragraphs between elements).

Alternatively, you can mix inquiry based learning (IBL) structure into a more traditionally formatted section using appropriate environments. You might start a section with an <activity>, for example.

You might want to break down the activity into smaller tasks, which can be done by specifying each <task>, as in the following project.

Of course you might still have definitions, theorems, and examples in a document like this.

And then you might have an example.

Examples are a good way to ask students to think about a problem but provide a solution right away. To encourage students to think about a problem more before giving the solution, you could use an exercise. In PreTeXt, exercises that occur inside a section, along with other content, are called “checkpoints”, even though they are coded simply as <exercise>

Of course, in this style of document, collecting important ideas is especially important.