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Let $A$ be the set of all undergraduate mathematical courses in the US and define a binary relation $\leq$ on $A$ such that for elements $a,b\in A$ (that is, $a$, $b$ are undergraduate mathematical courses), $a\leq b$ if and only if either $a=b$, or there exists finitely many courses $a_1,\ldots,a_n$ such that $a=a_1$, $b=a_n$ and $a_{i}$ is a prerequisite of $a_{i+1}$ for $i=1,\cdots, n-1$. For example, under the above definition, Pre-calculus $\leq$ Real Analysis and Linear Algebra $\leq$ Galois Theory. Also, Differential Equations and Point-set Topology are not comparable.

Under the definition above, it is easy to verify that $A$ is a finite partially ordered set. May I know where I can find a chart which gives the partial order structure in $A$?

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    $\begingroup$ Different schools arrange courses and prerequisites differently. $\endgroup$ – Henry Towsner Oct 30 '19 at 0:32
  • $\begingroup$ Yes, this question might be difficult to answer in full generality. It is only any given school that would have such a partial order, and even then there are different versions of each course at larger schools. For instance, at the same school some linear/matrix algebra courses might be both prerequisite to and after a proof introduction course. Instead, perhaps you could edit the question to ask for examples of typical partial orders. $\endgroup$ – kcrisman Oct 30 '19 at 1:21
  • $\begingroup$ Yes, it is probably the nature of proof expectations that will make this stray far from a partial order when combining multiple departments. For instance, when I went to Carnegie Mellon, combinatorics was a prereq for graph theory (usually given to seniors). The only reason for that is because CMU's graph theory expected strong proof skills and combo helped to build those skills. There's no reason graph theory couldn't be taught to freshmen at a different school as long as they didn't have the same expectation of proof maturity going in to it. $\endgroup$ – Matthew Daly Oct 30 '19 at 7:45
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The US has different courses at different schools. Sometimes with same name but differences in content or prereqs. It would be more meaningful to sketch this tree for a given school. Or do a few schools. That should give you some feel for the general lay of the land.

And I suggest to sketch it yourself, using a course catalog. You'll learn more doing a little work yourself. Also the course catalog will have course descriptions. Useful since same course may have different names at different schools (or visa versa). Not to say it is totally arbitrary of course. You'll see some stereotypical similarities.

Here is an example: https://www.undergradcatalog.registrar.vt.edu/1920/math.html Note that there are some remedial classes at the beginning which have no prereqs.

Also note that it is possible to have a "corequisite", which means that taking the coreq at same time is adequate (before is fine too). Or to have prereqs that are not just math courses (e.g. department chair permission or interview or placement exam or the like). Furthermore, it's less likely with math but certainly common for other departments to have prereqs that are out of department. IOW calculus before physics. Or physics before physical chemistry. Etc.

Here is another example: https://www.usna.edu/Academics/Majors-and-Courses/course-description/SM.php

You can select a few more on your own. Most schools have this stuff open and online nowadays.

Oh...and one more complexity: there are often a couple or more similar courses (e.g. enriched or specialized in some way) on the same content. So for the last link, if you look at SM 465 Advanced DiffyQs, it has a prereq of "(SM212 or SM222) [different versions of ODE courses] and (SM331 or SM331H or SM333). [different versions of real analysis courses]

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