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Introduction to the mathematics of discrete structures with emphasis on structures for computer science.
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<strong>UTSA Department of Mathematics</strong>
  
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To edit tables in each course below, you can use [https://tableconvert.com/mediawiki-to-excel MediaWiki-to-Excel converter] and/or the [https://tableconvert.com/excel-to-mediawiki Excel-to-MediaWiki converter]
  
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'''Catalog entry'''
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== Undergraduate Studies ==
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===STEM Core===
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* [[MAT1073]] College Algebra for Scientists and Engineers
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* [[MAT1093]] Precalculus
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* [[MAT1193]] Calculus for Biosciences
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* [[MAT1214]] Calculus I (4 credit hours)
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* [[MAT1213]] Calculus I (3 credit hours)
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* [[MAT1224]] Calculus II (4 credit hours)
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* [[MAT1223]] Calculus II (3 credit hours)
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* [[MAT2214]] Calculus III (4 credit hours)
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* [[MAT2213]] Calculus III (3 credit hours)
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* [[MAT2233]] Linear Algebra
  
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''Prerequisite'': Algebra and Number Systems (MAT 1313), or Discrete Mathematical Structures (CS 2233/2231), or instructor consent.
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===Minor in Mathematics===
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To receive a minor, students must complete at least 18 semester credit hours, including 6 hours at the upper-division level at UTSA, and must achieve a grade point average of at least 2.0 (on a 4.0 scale) on all work used to satisfy the requirements of a minor. See [https://catalog.utsa.edu/undergraduate/bachelorsdegreeregulations/minors/  UTSA's Undergraduate Catalog]
  
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''Contents'':
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The Minor in Mathematics requires the Calculus series plus linear algebra, and upper division courses in either the Math Major or the Data & Applied Science Core
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(1) Propositional logic: Axioms and Rules of Inference. Limitations of propositional logic: Informal introduction to quantifiers and syllogisms.
 
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(2) Predicate Logic: Existential and universal quantification, free variables and substitutions. Discussion of the various axiomatic systems for first-order logic (including axioms and rules of inference). The power and the limitations of axiomatic systems for logic: Informal discussion of the completeness and incompleteness theorems.
 
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(3) Sets and boolean algebras: Operations on sets. Correspondence between finitary set operations and propositional logic. Correspondence between infinitary operations and quantifiers. The power and limitations of the language of set theory: Informal discussion of the set-theoretic paradoxes and the need for axiomatic systems for set theory.
 
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(4) Relations: Special relations: Equivalence relations, partially ordered sets, maximum/minimum, maximal/minimal elements, least upper bounds and greatest lower bounds, totally ordered sets.
 
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(5) Functions: Operations of functions, direct image and inverse image.
 
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(6) Well-ordered sets: Correspondence between well-ordering relations and induction. Correspondence between well-ordering relations and choice functions.
 
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(7) Introduction to computability. Classical models of computation (recursive functions, and Turing models). Limitations of computation (the Halting Problem, fast-growing functions). Contemporary models of computation.
 
  
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===Data & Applied Science Core===
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==== Lower Division ====
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* [[MDC1213]] Sociocultural Foundations of Mathematics, Data Science, and Computing
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* [[MAT2253]] Applied Linear Algebra (3 credit hours)
  
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'''Sample textbooks''':
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==== Upper Division ====
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* [[MAT4133]]/[[MAT5133]] Mathematical Biology
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* [[MAT4143]]/[[MAT5143]] Mathematical Physics 1
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* [[MAT4XXX]]/[[MAT5XXX]] Mathematical Physics 2
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* [[MAT4XXX]]/[[MAT5XXX]] Introduction to Quantum Information Science and Engineering
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* [[MAT4373]]/[[MAT5373]] Mathematical Foundations of Statistics I (discrete & continuous PDFs)
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* [[MAT4383]]/[[MAT5383]] Mathematical Foundations of Statistics II (statistical inference)
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* [[MDC4153]]/[[MDC5153]] Mathematical Foundations of Data Analytics
  
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[1] Gordon Pace, ''Mathematics of Discrete Structures for Computer Science'', Springer, 2012
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===Math Major===
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==== Lower Division ====
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* [[MAT1313]] Algebra and Number Systems
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* [[MAT2313]] Combinatorics and Probability
  
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[2] Vladlen Koltun, ''Discrete Structures Lecture Notes, Stanford University'', 2008. Freely available [https://web.stanford.edu/class/cs103x/cs103x-notes.pdf here.]
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==== Upper Division ====
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* [[MAT3003]] Discrete Mathematics
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* <del>[[MAT3013]] Foundations of Mathematics</del> Course transitioning to be replaced by [[MAT3003]] Discrete Mathematics (below).
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* <del>[[MAT3213]] Foundations of Analysis</del> Course transitioning to be replaced by [[MAT3333]] Fundamentals of Analysis and Topology (below).
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* [[MAT3333]] Fundamentals of Analysis and Topology
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* [[MAT3233]] Modern Algebra
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* [[MAT3333]] Fundamentals of Analysis and Topology
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* [[MAT3313]] Logic and Computability
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* [[MAT3613]] Differential Equations I
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* [[MAT3623]] Differential Equations II
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* [[MAT3633]] Numerical Analysis
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* [[MAT3223]] Complex Variables
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* [[MAT4033]] Linear Algebra II
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* [[MAT4213]] Real Analysis I
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* [[MAT4223]] Real Analysis II
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* [[MAT4233]] Modern Abstract Algebra
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* [[MAT4273]] Topology
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* [[MAT4283]] Computing for Mathematics
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* [[MAT4323]] Applied Graph Theory
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* [[MAT4373]] Mathematical Statistics I
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* [[MAT43XX]]/[[MAT53XX]] Mathematics of Quantum Computing
  
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===Business===
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* [[MAT1053]] Algebra for Business
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* [[MAT1133]] Calculus for Business
  
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===Math for Liberal Arts===
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* [[MAT1043]] Introduction to Mathematics
  
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=== Elementary Education ===
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* [[MAT1023]] College Algebra
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* [[MAT1153]] Essential Elements in Mathematics I
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* [[MAT1163]] Essential Elements in Mathematics II
  
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=== General Math Studies===
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* [[MAT3233]] Modern Algebra
  
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== Graduate Studies ==
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=== Core Courses ===
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Core courses, common across all M.Sc. tracks, must be at least 50% of the credit 30 hours needed to obtain a M.Sc. degree. The following courses add up to 15 credit hours.
  
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* [[MAT5283]] Linear Algebra
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* [[MAT5253]] Real Analysis I
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* [[MAT5603]] Numerical Analysis
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* [[MAT5603]] Differential Equations I
  
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==Topics List==
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During completion of the required coursework, students must choose three distinct specialization tracks. For each of the three chosen tracks, the doctoral candidate must pass a qualifying examination that covers a two-semester course sequence approved for the given track. Courses with a numbered sequence I and II must be taken consecutively, and courses without a number can be taken in any order. Each track can be satisfied with any pair of these courses, unless otherwise noted.  Each track has a list of courses from which the two-semester course sequence for the examination must be chosen. Below is a list of approved qualifying tracks with their respective lists of courses:
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{| class="wikitable sortable"
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! Week !! Topic !! Sections from Pace's book !! !! Sections from Pace's book !! Prerequisites.
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=== Qualifying Exam Track: Algebra-Cryptography  ===
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|-
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|  1 
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* [[MAT 5173]] Abstract Algebra I
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|| [[Propositional logic]]
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* [[MAT 5183]] Abstract Algebra II
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|| 2.1-2.4
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* [[MAT 5123]] Introduction to Cryptography
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||
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* [[MAT 5243]] Cryptography II
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* Proofs
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* boolean models
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=== Qualifying Exam Track: Analysis ===
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* connections between boolean models and proofs
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|| MAT1313 or CS2233/2231, or equivalent.
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* [[MAT5203]] Real Analysis I
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|-
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* [[MAT5213]] Real Analysis II
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|  2 
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* [[MAT5223]] Complex Analysis I
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|| [[Completeness and soundness]]
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* [[MAT5233]] Complex Analysis II
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|| 2.5
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* [[MAT5553]] Harmonic Analysis
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* [[MAT5553]] Functional Analysis.
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* Completeness and soundness of propositional logic
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=== Qualifying Track: Mathematics of Information, Data and Communications  ===
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|-
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|  5-6 
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* [[MAT5123]] Introduction to Cryptography
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|| [[Predicate calculus]]
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* [[MAT5323]] Cryptography II
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|| 3.1-3.5
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* [[MAT5433]] Graph Theory (Discrete Mathematics II)
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* [[MAT5153]] Data Analytics
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* Limits of propositional logic
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* [[MAT5XXX]] Quantum Information
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* free variables and substitution.
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* [[MAT5ZZZ]] Mathematics of AI
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||
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|-
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=== Qualifying Exam Track: Differential Equations ===
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|  7 
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|| [[Sets and boolean algebras]]
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* [[MAT5653]] Differential Equations I
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|| 4.1-4.5
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* [[MAT5673]] Partial Differential Equations I
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* [[MAT5683]] Partial Differential Equations II
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* Set comprehension.
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* Finitary and general operations on sets.
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=== Qualifying Exam Track: Scientific Computing ===
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||
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|-
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* [[MAT5603]] Numerical Analysis
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|  8 
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* [[MAT5293]] Numerical Linear Algebra
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|| [[Sets and boolean algebras]]
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* [[MAT5613]] Numerical Solutions of Differential Equations
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|| 4.6
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* [[MAT5163]] Probability and Computing
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||
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* Boolean algebras and boolean rings.
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=== Qualifying Exam Track: Discrete Mathematics and Logic ===
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||
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|-
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* [[MAT5423]] Discrete Mathematics
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|  9 
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* [[MAT5433]] Discrete Mathematics II (Graph Theory)
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|| [[Relations]]  
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* [[MAT5443]] Logic and Computability
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|| 5.1-5.7
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* [[MAT5453]] Quantum Logic
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||
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* Relations and sets
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=== Qualifying Exam Track: Topology and Geometry ===
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* Inverse of a relation and composition of relations
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* Beyond binary relations
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* [[MAT5243]] Topology I
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||
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* [[MAT5253]] Topology II
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|-
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* [[MAT5253]] Differential Geometry
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|  10 
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|| [[Classifying Relations]]  
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=== Qualifying Exam Track: Mathematical Biology  ===
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|| 6.1-6.3
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||
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* [[MAT5133]] Mathematical Biology
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* Totality
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* Surjectivity
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Plus one of the following two courses:
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* Injectivity
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* Functionality
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* [[MAT5323]] Mathematical Modeling
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||
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* Optimal Control and Calculus of Variations*.
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|-
 
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|  11-12 
 
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|| [[Discrete structures]]  
 
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|| 7.1-8.4
 
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||
 
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* Graphs
 
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* Semigroups
 
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* groups
 
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||
 
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|-
 
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|  13-14 
 
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|| [[Reasoning about programs]]  
 
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|| 10.1-10.4
 
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||
 
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* Algorithms
 
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* Program semantics
 
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* Uncomputability
 
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||
 
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|}
 

Revision as of 15:44, 17 August 2025

UTSA Department of Mathematics

To edit tables in each course below, you can use MediaWiki-to-Excel converter and/or the Excel-to-MediaWiki converter

Undergraduate Studies

STEM Core

  • MAT1073 College Algebra for Scientists and Engineers
  • MAT1093 Precalculus
  • MAT1193 Calculus for Biosciences
  • MAT1214 Calculus I (4 credit hours)
  • MAT1213 Calculus I (3 credit hours)
  • MAT1224 Calculus II (4 credit hours)
  • MAT1223 Calculus II (3 credit hours)
  • MAT2214 Calculus III (4 credit hours)
  • MAT2213 Calculus III (3 credit hours)
  • MAT2233 Linear Algebra

Minor in Mathematics

To receive a minor, students must complete at least 18 semester credit hours, including 6 hours at the upper-division level at UTSA, and must achieve a grade point average of at least 2.0 (on a 4.0 scale) on all work used to satisfy the requirements of a minor. See UTSA's Undergraduate Catalog

The Minor in Mathematics requires the Calculus series plus linear algebra, and upper division courses in either the Math Major or the Data & Applied Science Core

Data & Applied Science Core

Lower Division

  • MDC1213 Sociocultural Foundations of Mathematics, Data Science, and Computing
  • MAT2253 Applied Linear Algebra (3 credit hours)

Upper Division

Math Major

Lower Division

  • MAT1313 Algebra and Number Systems
  • MAT2313 Combinatorics and Probability

Upper Division

Business

Math for Liberal Arts

  • MAT1043 Introduction to Mathematics

Elementary Education

  • MAT1023 College Algebra
  • MAT1153 Essential Elements in Mathematics I
  • MAT1163 Essential Elements in Mathematics II

General Math Studies

Graduate Studies

Core Courses

Core courses, common across all M.Sc. tracks, must be at least 50% of the credit 30 hours needed to obtain a M.Sc. degree. The following courses add up to 15 credit hours.

During completion of the required coursework, students must choose three distinct specialization tracks. For each of the three chosen tracks, the doctoral candidate must pass a qualifying examination that covers a two-semester course sequence approved for the given track. Courses with a numbered sequence I and II must be taken consecutively, and courses without a number can be taken in any order. Each track can be satisfied with any pair of these courses, unless otherwise noted. Each track has a list of courses from which the two-semester course sequence for the examination must be chosen. Below is a list of approved qualifying tracks with their respective lists of courses:

Qualifying Exam Track: Algebra-Cryptography

Qualifying Exam Track: Analysis

Qualifying Track: Mathematics of Information, Data and Communications

Qualifying Exam Track: Differential Equations

  • MAT5653 Differential Equations I
  • MAT5673 Partial Differential Equations I
  • MAT5683 Partial Differential Equations II

Qualifying Exam Track: Scientific Computing

  • MAT5603 Numerical Analysis
  • MAT5293 Numerical Linear Algebra
  • MAT5613 Numerical Solutions of Differential Equations
  • MAT5163 Probability and Computing

Qualifying Exam Track: Discrete Mathematics and Logic

Qualifying Exam Track: Topology and Geometry

Qualifying Exam Track: Mathematical Biology

Plus one of the following two courses:

  • MAT5323 Mathematical Modeling
  • Optimal Control and Calculus of Variations*.
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