## Placement Overview

### Placement Test Links

### Understanding Your Math Placement

Each student who takes the UW math placement test is assigned a placement group via our algorithm. When you view your placement test scores, you’ll see one of the groups below assigned to you. Each group qualifies you for a specific set of math courses, and also lets you know whether you satisfied or need to complete QR-A.

Below is a general overview of the math courses which you qualify for via the placement test. **If you’re a new student attending SOAR, you’ll have a conversation with an academic advisor about which course best fits into your academic plan. **We also have some more info for new students here.

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### MATH 96 (this course does not count for degree credit)

### Precalculus

- MATH 96

### MATH 96 (this course does not count for degree credit) or MATH 141. Must take MATH 96 if additional math courses are required (for major or as prerequisite to other courses)

### Precalculus

- MATH 96

### Non-Precalculus QR-A

- MATH 141 (or other non-precalculus QR-A course)

### MATH 112 (followed by MATH 113 for MATH 221)

### Precalculus

- MATH 112

### Non-Precalculus QR-A

- MATH 141 (or other non-precalculus QR-A course)

### MATH 112 (will not need MATH 113 for MATH 221) or MATH 114 or MATH 171/217 sequence (must take both courses and is equivalent to MATH 114 and MATH 221)

### Precalculus

- MATH 112 (will not need MATH 113 if moving on to MATH 221)
- MATH 114
- MATH 171/217 sequence

### Non-Precalculus QR-A

- MATH 141 (or other non-precalculus QR-A course)

### MATH 114 or MATH 112 (followed by MATH 113 for MATH 221) or MATH 171/217 sequence (must take both courses and is equivalent to MATH 114 and MATH 221). [QR-A satisfied]

### Precalculus

- MATH 112
- MATH 114
- MATH 171/217 sequence

### Non-Precalculus QR-A

- MATH 141 (or other non-precalculus QR-A course)

*QR-A is satisfied via this placement.*

### MATH 112 (will not need MATH 113 for MATH 221) or MATH 114 or MATH 171/217 sequence (must take both courses and is equivalent to MATH 114 and MATH 221) [QR-A satisfied]

### Precalculus

- MATH 112 (will not need MATH 113 if moving on to MATH 221)
- MATH 114
- MATH 171/217 sequence

### Non-Precalculus QR-A

- MATH 141 (or other non-precalculus QR-A course)

*QR-A is satisfied via this placement.*

### MATH 113 (will not need MATH 112 for MATH 221) or MATH 211 or MATH 114 or MATH 171/217 sequence (must take both courses and is equivalent to MATH 114 and MATH 221). [QR-A satisfied]

### Precalculus

- MATH 113 (will not need MATH 112 if moving on to MATH 221)
- MATH 114
- MATH 171/217 sequence

### Calculus

- MATH 211

### Non-Precalculus QR-A

- MATH 141 (or other non-precalculus QR-A course)

*QR-A is satisfied via this placement.*

### MATH 211 or MATH 221 [QR-A satisfied]

### Calculus

- MATH 211
- MATH 221

### Non-Precalculus QR-A

- MATH 141 (or other non-precalculus QR-A course)

*QR-A is satisfied via this placement.*

### A Note About MATH 141

MATH 141 is a terminal, QR-A math course that does not lead to enrollment into higher-level math classes, and its completion does not imply preparedness to move on with our precalculus coursework. However, if you are very confident that you just need a QR-A course, MATH 141 is one option, and it may be a very good fit as it not necessarily feel like other math classes you may have taken.

Due to the nature of the course and student placement, students should connect with their advisor before choosing this course.

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### Early Sequencing Roadmap

## Precalculus

### Precalculus Links

### Precalculus Course Overviews

Below is an overview of our precalculus courses, as well as some considerations on which students these courses may be best for.

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### MATH 96 (Preparatory Algebra)

### What is MATH 96?

MATH 96 is a developmental math course that is intended to prepare students for further math courses or, in some cases, QR-A coursework. Some students may or may not need to take MATH 96 depending on their particular placement, so having a conversation with their academic advisor is important to determine whether the course is necessary or recommended.

### Who should take MATH 96?

MATH 96 is great for students who…

- are required to take the course to satisfy math remediation and/or qualify for QR-A enrollment.
- are interested in continuing on to further precalculus coursework
*or if there is the possibility of wanting/needing to enroll in precalculus in the future.***Note:**MATH 114 is*not*suggested as the course to take after MATH 96. We strongly suggest that students in MATH 96 continue to MATH 112 instead.

### MATH 112 (Algebra)

### What is MATH 112?

MATH 112 is the first half of our two-semester precalculus sequence: MATH 112 and MATH 113.

MATH 112 covers algebra, which prepares students to move into trigonometry (MATH 113) or the survey of calculus **without** trigonometry (MATH 211).

### Who should take MATH 112?

MATH 112 is great for students who need algebra, but…

- do not need the trigonometry experience which MATH 114 or MATH 171/217 provides.
- would like to opt for a slower-paced precalculus sequence by splitting algebra and trigonometry into two semesters, the alternatives being:
- MATH 114, an accelerated and fast-paced precalculus which combines topics from MATH 112 and MATH 113 into one semester.
- MATH 171/217, a faster-paced, two-semester sequence which combines “just in time” review of algebra and trigonometry with calculus topics.

We recommend students consider MATH 112 if they are not comfortable with the accelerated nature of MATH 114 and the MATH 171/217 sequence–both of which are 5 credits per semester–or if they would like to balance their course schedules out more.

Students can also consider MATH 112 if they are planning on moving into MATH 211 instead of MATH 221.

### MATH 113 (Trigonometry)

### What is MATH 113?

MATH 113 is the second half of our two-semester precalculus sequence: MATH 112 and MATH 113.

MATH 113 covers trigonometry, which prepares students to move into calculus with trigonometry (MATH 221). Trigonometry is **not** required for students who intend to take MATH 211 for calculus.

### Who should take MATH 113?

MATH 113 is great for students who need trigonometry, but…

- do not need the algebra experience which MATH 114 or MATH 171/217 provides.
- would like to opt for a slower-paced trigonometry course, the alternatives being:
- MATH 114, an accelerated and fast-paced precalculus which combines topics from MATH 112 and MATH 113 into one semester.
- MATH 171/217, a faster-paced, two-semester sequence which combines “just in time” review of algebra and trigonometry with calculus topics.

We recommend students consider MATH 113 if they are not comfortable with the accelerated nature of MATH 114 and the MATH 171/217 sequence–both of which are 5 credits per semester–or if they would like to balance their course schedules out more.

### MATH 114 (Algebra and Trigonometry)

### What is MATH 114?

MATH 114 is an accelerated, fast-paced precalculus course that combines topics from MATH 112 (algebra) and MATH 113 (trigonometry) into one semester.

MATH 114 covers algebra and trigonometry, which prepares students to move into calculus with trigonometry (MATH 221).

### Who should take MATH 114?

MATH 114 is great for students who…

- are comfortable with the accelerated nature of the course and are confident that they can keep up with the pacing.
- would like to opt for a dedicated precalculus course before moving into MATH 221, as opposed to MATH 171/217 which is a fast-paced two-semester sequence that combines “just in time” review of algebra and trigonometry with calculus topics.

If a student intends on enrolling in MATH 114, they should be aware that the course is **fast-paced** and **accelerated** in nature. The course is 5 credits and will require a good amount of time and focus to keep up with the pacing and be successful. If they do not wish to take a course with this fast of a pace, they should consider the MATH 112/113 sequence instead.

MATH 114 is also an option for students who are still deciding on which calculus pathway they wish to pursue, as MATH 114 can be followed by either MATH 211 or MATH 221.

### MATH 171/217 (Calculus with Algebra and Trigonometry I & II)

### What is MATH 171/217?

MATH 171/217 is a fast-paced, two-semester sequence that combines “just in time” review of algebra and trigonometry (precalculus) with calculus topics.

Students who complete MATH 171/217 are considered to have completed the equivalent of precalculus ([MATH 112 and 113] or MATH 114) and first-semester calculus with trigonometry (MATH 221).

### Who should take MATH 171/217?

MATH 171/217 is great for students who…

- are comfortable with the accelerated nature of the sequence and are confident that they can keep up with the pacing.
- are interested in a course sequence which reviews precalculus topics and also gives more immediate exposure to calculus.

If a student intends on enrolling in MATH 171/217, they should be aware that the course sequence is **fast-paced** and **accelerated** in nature. The courses are 5 credits and will require a good amount of time and focus to keep up with the pacing and be successful. If they do not wish to take a course sequence with this fast of a pace, they should consider the MATH 112/113 sequence instead.

### Important Reminders

- MATH 171 is only offered in the fall, and MATH 217 is only offered in the Spring.
- Students must pass both courses in order to be awarded credit for the equivalent of MATH 114 and MATH 221,
**but MATH 171 does not equal MATH 114, nor does MATH 217 equal MATH 221.**- Note that completing MATH 171 but not MATH 217 still awards course credit, but MATH 171 alone does not qualify a student for any math course outside of MATH 217.

- If a student takes MATH 171 and does not take MATH 217, and then decides to take MATH 221, they must take the prerequisite coursework for MATH 221.

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### [MATH 112 and/or MATH 113] vs. MATH 114 vs. MATH 171/217

Students deciding between [MATH 112 and/or MATH 113], MATH 114, and MATH 171/217 should keep the following things in mind:

- MATH 171/217 is typically suggested for students in technical majors, like physics or engineering, or students who have already seen calculus before.
- MATH 114 provides more flexibility than MATH 171/217 (MATH 114 and 211/221 are offered every fall/spring semester), though is still
**fast-paced**and**accelerated**in nature. - Students can consider MATH 112 and/or MATH 113 (depending on their placement) if they would like a slower-paced version of precalculus, as opposed to MATH 114 or the MATH 171/217 sequence, both of which are more accelerated in nature.

## Calculus

### Calculus Course Overviews

Below is an overview of our calculus courses, as well as some considerations on which students these courses may be best for.

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### MATH 211 (Calculus)

### What is MATH 211?

MATH 211 is a first-semester survey of calculus course *without* trigonometry.

**MATH 211 is only a prerequisite for MATH 213, and no other math courses. **

- Students
**cannot**take the MATH 211/213 course sequence and then continue into the “main” calculus sequence (MATH 222 and MATH 234). For this reason, students should take MATH 221 if there is even the*slightest*chance they will require or want further mathematics courses. - Students who take MATH 211 and change their mind later will be required to go back to MATH 221. Students who complete MATH 211 but did not place directly into MATH 221 will need to go back and take the MATH 221 prerequisite coursework or retake the placement test.
**MATH 211 is not a prerequisite to MATH 221.**

MATH 211 is primarily taken by students whose majors or programs of interest do **not** require MATH 221 and will accept MATH 211—such as some programs in the Wisconsin School of Business and the College of Agricultural and Life Sciences—or who are certain do not need MATH 221 to meet course prerequisites (like those for MATH 222, MATH/COMP SCI 240, PHYSICS 201 or 207, etc.).

### Who should take MATH 211?

MATH 211 is great for students who…

- would like to take a calculus course but do
**not**need MATH 221 for satisfying course prerequisites or major requirements. - are
**not**planning on taking the “main” calculus sequence (MATH 221, 222, or 234) or further math coursework (like MATH 340) besides MATH 213. - have
**had a conversation with their academic advisor**and have determined that MATH 211 is satisfactory for their needs.

### Important Considerations

Pre-business students (those looking to apply to the Wisconsin School of Business but as of yet are not admitted) should **strongly **consider MATH 221 as an alternative to MATH 211. MATH 211 has less utility outside of the Wisconsin School of Business as it is often **not **accepted as an alternative course to MATH 221 for program requirements and course prerequisites. MATH 211 is also **not **an acceptable prerequisite course for MATH 221, and if a student takes MATH 211 they may need to complete MATH 221’s prerequisite coursework (depending on placement).

**Students should have a conversation with their academic advisor if they are uncertain about which course is most appropriate for them.**

### MATH 213 (Calculus and Introduction to Differential Equations)

### What is MATH 213?

MATH 213 is a second survey-level calculus course designed to follow MATH 211. The course covers further topics in calculus, including some topics from differential equations. As with MATH 211, this course does not use trigonometric functions.

### Who should take MATH 213?

This course is primarily designed for finance majors in the Wisconsin School of Business.

### MATH 221 (Calculus and Analytic Geometry 1)

### What is MATH 221?

MATH 221 is a first-semester calculus course *with *trigonometric functions. It covers differential calculus and the beginning of integral calculus.

MATH 221 is the most commonly taken calculus course at UW-Madison, and is accepted as a prerequisite for MATH 222, which then leads to the rest of the mathematics curriculum (MATH 234, 340, etc.).

### Who should take MATH 221?

MATH 221 is great for students who…

- are interested in a widely-applicable and flexible first-semester calculus course which satisfies many course prerequisites or program requirements that require calculus.
- are interested in pursuing further math coursework, such as MATH 222, MATH 234, MATH 319, MATH 340, etc.

### MATH 222 (Calculus and Analytic Geometry 2)

### What is MATH 222?

MATH 222 is a second-semester calculus course *with* trigonometric functions. It covers techniques of integration, sequences, and series. It also includes a brief introduction to differential equations and a brief introduction to vectors.

### Who should take MATH 222?

MATH 222 is great for students who…

- are interested in a widely-applicable and flexible second-semester calculus course which satisfies many course prerequisites or program requirements that require calculus.
- are interested in pursuing further math coursework, such as MATH 234, MATH 319, MATH 340, etc.

### MATH 234 (Calculus--Functions of Several Variables)

### What is MATH 234?

MATH 234 is a third-semester calculus course. It is the study of multivariable functions, including an introduction to vectors and concluding with topics in vector calculus that are essential in physics and certain areas of engineering.

### Who should take MATH 234?

MATH 234 is great for students who…

- are interested in studying calculus of more than one variable, which is frequently required by course prerequisites or program requirements that require calculus.
- are interested in pursuing more advanced mathematics courses, including possibly majoring in mathematics.

### MATH 375/376 (Topics in Multi-Variable Calculus & Differential Equations)

### What is MATH 375/376?

MATH 375/376 is a year-long, rigorous, **proofs-based** honors calculus sequence.

MATH 375/376 serves as a more theoretical and advanced alternative to multi-variable calculus (MATH 234), differential equations (MATH 319), and linear algebra (MATH 341). The goal of these courses is to provide highly motivated and well-prepared students with an opportunity to develop a deeper understanding of these fundamental areas of mathematics.

### Who should take MATH 375/376?

MATH 375/376 is great for students who…

- are looking for a rigorous challenge.
- do
**not**have credit for any of MATH 234, 319, 320, 340, or 341 (due to content overlap). - are interested in pursuing higher-level mathematics coursework for which proofs experience is required.

The Honors Calculus: MATH 375/376 page has more detailed information on enrollment requirements, considerations, FAQs, and how to enroll.

### MATH 228 (WES-Calculus)

WES-Calculus sections are 2-credit supplementary discussion sections that students can apply for and take along with a calculus course.

WES-Calculus sections meet for more time than non-WES calculus discussion sections.

## Beyond Calculus

### Beyond Calculus Links

### Linear Algebra

UW-Madison offers four distinct introductory-level algebra courses.

In order to complete the major in mathematics you must take a course in linear algebra. At UW-Madison, we offer several versions of linear algebra. **Note that in all versions of the major and certificate, only one of the following courses may be used to fulfill any of the requirements.**

The purpose of this page is to describe the essential differences between the four introductory-level linear algebra courses.

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### MATH 320 (Linear Algebra and Differential Equations)

MATH 320 covers both some linear algebra and some differential equations. As such, students who complete this course can consider themselves as also having some of the content of MATH 319 (Techniques in Ordinary Differential Equations). This course lends itself to the MATH 321-322 applied analysis sequence.

Students who have completed MATH 320 will need to complete one of MATH 421; the applied analysis sequence MATH 321-322; or MATH 467 before moving on to 500-level courses that require introductory proofs coursework.

In a regular fall or spring semester, MATH 320 is often taught with both general sections and an Accelerated Honors (!) section. There are some differences between the two:

### General MATH 320

Differential equations and linear algebra are crucial for the analysis and control of many dynamical systems such as electrical networks, mechanical structures, chemical and biological networks. Those applications lead to often very large systems of coupled differential equations. The study of the equilibria, stability and control of such systems requires the concepts and tools of differential equations and linear algebra. MATH 320 is a first course in the study of differential equations and linear algebra.

### MATH 320 Honors

Accelerated Honors at UW-Madison are Honors courses open to anyone and **not** restricted to students in the L&S Honors program. MATH 320 Honors is recommended for students in all applied areas including all areas of engineering, as well as the physical, chemical, and biological sciences. Differential equations and linear algebra are crucial to the study of dynamical systems such as electrical networks, mechanical structures, and chemical reaction networks. MATH 320 Honors has recently used Gilbert Strang’s *Differential Equations and Linear Algebra* textbook. MATH 320 Honors covers some material that is important for engineering and data science applications but may not be covered in the general (non-Honors) MATH 320 lectures such as, for example, the Dirac delta function and impulse response, the PLU and CR matrix decompositions, together with an introduction to the use of modern computational tools such as MATLAB (or Python).

In summary, MATH 320…

- Is useful for students interested in classical applications of mathematics (e.g., physics, engineering, continuous modeling, etc.);
- Covers material in MATH 319 and therefore
**credit for only one of MATH 319 or 320 can be applied to the math major or certificate;** - Covers material in MATH 340 and therefore
**credit for only one of MATH 320 or 340 can be applied to the math major or certificate;** - Is not by itself sufficient for taking advanced math courses;
- Is offered with an Accelerated Honors (!) version in some semesters. This version is suggested for potential math majors and those in the AMEP program.

Suggested further courses are…

- The applied analysis sequence MATH 321-322, which covers more mathematics useful for traditional applications;
- MATH 415, which includes both continuous and discrete models of changing systems;
- MATH 421 for an introduction to more formal mathematical arguments.

### MATH 340 (Elementary Matrix and Linear Algebra)

MATH 340 is a basic linear algebra course which focuses on vectors as ordered sets of real numbers and linear operators as matrices. In this course the focus is typically on computational aspects of the subject with some lighter treatment of the more theoretical points.

Students who complete this course and would also like exposure to differential equations should consider MATH 319.

Students who have completed MATH 340 will need to complete one of MATH 421; the applied analysis sequence MATH 321-322; or MATH 467 before moving on to 500-level courses that require introductory proofs coursework.

In summary, MATH 340…

- Is ideal for students who need comprehensive functional knowledge of basic matrix algebra, and in particular those looking for applications featuring discrete mathematics (i.e., computer science, data science, statistics, and industrial and systems engineering);
- Is not by itself sufficient for enrollment in advanced math courses.
- Covers material in MATH 320 and therefore
**credit for only one of MATH 320 or 340 can be applied to the math major or certificate;** - Covers material in MATH 341 and therefore
**credit for only one of MATH 340 or 341 can be applied to the math major or certificate;**

Suggested further courses are…

- MATH 319 for those interested in the applied analysis sequence;
- MATH 421 for those interested in advanced undergraduate math courses above the 500-level.

### MATH 341 (Linear Algebra)

MATH 341 is a linear algebra course which is also meant to be an introduction to proofs and proof-writing. The linear algebra content of the course is more robust than any of the others listed on this page. Students who complete the course should be well-prepared to move on to any upper-level course, in particular MATH 521, 541, or 551.

It is the recommended linear algebra course for majors interested in moving to advanced undergraduate courses quickly.

**Due to the more intensive proof-writing nature of the course, students will probably find the course more demanding than MATH 340, and for this reason MATH 341 carries the Accelerated Honors (!) label.**

Students who complete this course and would also like exposure to differential equations should consider MATH 319.

In summary, MATH 341…

- Is accepted in both the major and certificate programs;
- Is a good introduction to proofs and proof-writing;
- Covers material in MATH 320 and therefore
**credit for only one of MATH 320 or 341 can be applied to the math major or certificate;** - Covers material in MATH 340 and therefore
**credit for only one of MATH 340 or 341 can be applied to the math major or certificate;** - Will give students access to advanced-level undergraduate math courses.

Suggested further courses are…

- MATH 421 for another exposure to formal mathematical arguments at the introductory-level;
- Any math course above the 500-level (possibly assuming other prerequisites).

### MATH 375 (Topics in Multi-Variable Calculus and Linear Algebra)

MATH 375 is an Accelerated Honors (!) course which features the role that linear algebra has in multivariable calculus.

It is assumed that students who complete this course will move on to complete the sequel course, MATH 376. (Students who complete MATH 375 and not MATH 376 are not considered to have completed the content of MATH 234! So by enrolling in MATH 375 in the fall, you should be prepared to enroll in MATH 376 in the spring, or MATH 234 in order to complete multivariate calculus.)

In summary, MATH 375:

**Is Honors-level;**- Enrollment is by permission only;
- Is not a course you can take if you have credit for one or more of MATH 234, 319, 320, 340, or 341;
- Is a good introduction to proofs and proof-writing.

More information on the MATH 375/376 sequence can be found on our Honors calculus page.

### Avoiding Duplicated Coursework

Within the mathematics major (including all named options) and the mathematics certificate, there are three course groups from which a student may count **at most one** class. These are:

- Introduction to Linear Algebra (MATH 320, 340, 341, 375)
- Introduction to Differential Equations (MATH 319, 320, 376)
- Introduction to Probability (MATH 309, 331, 431)

If you have a question that is not answered here, you may email the Associate Director of Undergraduate Studies, Dr. Mitch Keller.

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### I have credit for MATH 319.

If you already have credit for MATH 319, then you should not take MATH 320 or MATH 376.

- If you want
*linear algebra*content, we recommend MATH 340 or MATH 341.- You will want MATH 341 if you need an introduction to
*writing formal proofs*as required in the mathematics major, but otherwise MATH 340 is a good choice.

- You will want MATH 341 if you need an introduction to
- If you want more content in
*differential equations*, consider MATH 415 or MATH 519. MATH 320 will cover fewer topics in differential equations that what you learned in MATH 319.

### I have credit for MATH 320.

MATH 320 provided you with an introduction to both linear algebra and differential equations. Thus, you should **not** take MATH 319, 340, 341, 375, or 376.

- If you want to learn more
*linear algebra*, consider MATH 443. - If you want to learn more
*differential equations*, consider MATH 415 or 519. - If you want to to prepare for
*proof-based mathematics coursework*, take MATH 421.

The Applied Mathematical Analysis sequence (MATH 321-322) is a nice follow-up to MATH 320. MATH 322 includes additional topics in differential equations.

### I have credit for MATH 340.

MATH 340 is a comprehensive course in linear algebra. Thus, you should not take MATH 320, 341, or 375.

- If you would like to learn
*differential equations*, take MATH 319. - If you need to satisfy the
*proofs*prerequisite for more advanced mathematics courses, take MATH 421 or MATH 467. - If you would like to learn more
*linear algebra*, consider MATH 443.

### I have credit for MATH 309 or MATH 431.

The mathematics and statistics departments consider STAT/MATH 309 and MATH/STAT 431 interchangeable, so no student should take both.

If you would like a more theoretical course in probability, consider MATH 531. For a follow-up course to your introduction to probability, consider MATH 632.

### I have credit for MATH 375/376.

MATH 375-376 covers, from the Math Department’s perspective, the content of MATH 234, 319, and 341. If you have completed or are taking this sequence, you should **not** take MATH 234, 319, 320, 340, or 341. If you have completed MATH 375 and want an additional math class to take alongside MATH 376, you might consider MATH 443, which covers additional linear algebra, or MATH 475, which provides an introduction to combinatorics.

### I am enrolled in more than one of these classes.

If you want to take more than one of these classes (other than MATH 375/376, which is addressed above), here are some recommendations:

- MATH 319 and MATH 340 introduce both
*differential equations*and*linear algebra*. This is preferred to taking MATH 319 and MATH 320 or taking MATH 320 and MATH 340. - MATH 319 and MATH 341 introduce
*differential equations*,*linear algebra*, and*writing formal mathematical proofs*. If you are interested in taking more advanced (500-level and above) mathematics courses, this could be a good option.- MATH 320 and MATH 421 would be another combination to learn
*differential equations*,*linear algebra*, and*writing formal mathematical proofs*.

- MATH 320 and MATH 421 would be another combination to learn