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<strong>UTSA Department of Mathematics</strong>
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Introduction to the mathematics of discrete structures with emphasis on structures for computer science.
  
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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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== Undergraduate Studies ==
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'''Catalog entry'''
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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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* [[MAT1224]] Calculus II (4 credit hours)
 
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* [[MAT2214]] Calculus III (4 credit hours)
 
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* [[MAT2233]] Linear Algebra
 
  
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===Data & Applied Science Core===
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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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* [[MDC1213]] Mathematics, Data, AI and the Modern World
 
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* [[MAT1213]] Calculus I (3 credit hours)
 
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* [[MAT1223]] Calculus II (3 credit hours)
 
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* [[MAT2213]] Calculus III (3 credit hours)
 
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* [[MAT2243]] Applied Linear Algebra (3 credit hours)
 
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* [[MAT4133]]/[[MAT5133]] Mathematical Biology
 
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* [[MAT4143]]/[[MAT5143]] Mathematical Physics
 
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* [[MAT4373]]/[[MAT5373]] Mathematical Statistics I (discrete & continuous PDFs)
 
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* [[MAT4383]]/[[MAT5383]] Mathematical Statistics II (statistical inference)
 
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* [[MDC4413]] Data Analytics
 
  
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===Math Major===
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''Contents'':
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* [[MAT1313]] Algebra and Number Systems
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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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* [[MAT2313]] Combinatorics and Probability
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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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* [[MAT3003]] Discrete Mathematics
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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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* <s>[[MAT3013]] Foundations of Mathematics </s> Replaced by MAT3003 Discrete Mathematics
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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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* [[MAT3213]] Foundations of Analysis
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(5) Functions: Operations of functions, direct image and inverse image.
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* [[MAT3233]] Abstract Algebra I
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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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* [[MAT3613]] Differential Equations I
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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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* [[MAT3623]] Differential Equations II
 
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* [[MAT3633]] Numerical Analysis
 
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* [[MAT3223]] Complex Variables
 
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* [[MAT4213]] Real Analysis I
 
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* [[MAT4223]] Real Analysis II
 
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* [[MAT4233]] Abstract Algebra II
 
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* [[MAT4273]] Topology
 
  
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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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'''Sample textbooks''':
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* [[MAT1043]] Introduction to Mathematics
 
  
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=== Elementary Education ===
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[1] Gordon Pace, ''Mathematics of Discrete Structures foe Computer Science'', Springer, 2012
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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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[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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* [[MAT3233]] Modern Algebra
 
  
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== Graduate Studies ==
 
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=== Core M.Sc. Studies ===
 
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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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* Two courses in the Analysis and Algebra sequences in the following combinations:
 
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** [[MAT5173]] Algebra  I & [[MAT5183]] Algebra II.
 
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** [[MAT5203]] Analysis I  & [[MAT5213]] Analysis II
 
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** [[MAT5173]] Algebra &  [[MAT5203]] Analysis I.
 
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* [[MAT5283]] Linear Algebra (fall odd years)
 
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* [[MAT5423]] Discrete Mathematics I (fall even years)
 
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* [[MAT4373]]/[[MAT5373]] Mathematical Statistics I  (fall even years)
 
  
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=== Qualifying Examination Tracks  ===
 
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* [[MAT5183]] Algebra II (fall, even years) (Pure, Applied tracks)
 
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* [[MAT5123]] Cryptography (spring even years) (Pure, Applied tracks)
 
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* [[MAT5323]] Cryptography II (spring odd years) (Pure, Applied tracks))
 
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* [[MAT5213]] Analysis II (spring even years)  (Pure track)
 
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* [[MAT5113]] Computing for Mathematics (spring even years)  (Pure, Applied tracks)
 
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* [[MAT5433]] Discrete Mathematics II (spring odd years)  (Pure, Applied tracks)
 
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* [[MAT4383]]/[[MAT5383]] Mathematical Statistics II  (fall even years)  (Applied tracks)
 
  
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=== M.Sc. Track in Pure Mathematics  ===
 
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* [[MAT4423/MAT5443]] Logic and Computability
 
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* [[MAT5243]] General Topology
 
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* [[MAT5253]] General Topology II
 
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* [[MAT5323]] Cryptography II
 
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* [[MAT5183]] Algebra II
 
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* [[MAT5223]] Theory of Functions of a Complex Variable
 
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* [[MAT5343]] Differential Geometry
 
  
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=== M.Sc. Track in Applied & Industrial Mathematics ===
 
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* [[MAT4153]]/[[MAT5153]] Data Analytics
 
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* [[AIM 5113]] Introduction to Industrial Mathematics
 
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* [[MAT 5113]] Computing for Mathematics
 
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* [[MAT 5653]] Differential Equations I
 
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* [[MAT 5673]] Partial Differential Equations
 
  
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=== M.Sc. in Mathematics Education ===
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==Topics List==
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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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|-
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|  1 
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|| [[Propositional logic]]
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|| 2.1-2.4
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||
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* Proofs
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* boolean models
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* connections between boolean models and proofs
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|| MAT1313 or CS2233/2231, or equivalent.
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|-
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|  2 
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|| [[Completeness and soundness]]
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|| 2.5
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||
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* Completeness and soundness of propositional logic
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||
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|-
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|  5-6 
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|| [[Predicate calculus]]
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|| 3.1-3.5
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||
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* Limits of propositional logic
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* free variables and substitution.
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||
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|-
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|  7 
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|| [[Sets and boolean algebras]]
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|| 4.1-4.5
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||
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* Set comprehension.
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* Finitary and general operations on sets.
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||
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|-
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|  8 
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|| [[Sets and boolean algebras]]
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|| 4.6
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||
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* Boolean algebras and boolean rings.
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||
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|-
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|  9 
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|| [[Relations]]
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|| 5.1-5.7
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||
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* Relations and sets
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* Inverse of a relation and composition of relations
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* Beyond binary relations
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||
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|-
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|  10 
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|| [[Classifying Relations]]
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|| 6.1-6.3
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||
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* Totality
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* Surjectivity
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* Injectivity
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* Functionality
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||
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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 16:42, 24 March 2023

Introduction to the mathematics of discrete structures with emphasis on structures for computer science.


Catalog entry

Prerequisite: Algebra and Number Systems (MAT 1313), or Discrete Mathematical Structures (CS 2233/2231), or instructor consent.

Contents: (1) Propositional logic: Axioms and Rules of Inference. Limitations of propositional logic: Informal introduction to quantifiers and syllogisms. (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. (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. (4) Relations: Special relations: Equivalence relations, partially ordered sets, maximum/minimum, maximal/minimal elements, least upper bounds and greatest lower bounds, totally ordered sets. (5) Functions: Operations of functions, direct image and inverse image. (6) Well-ordered sets: Correspondence between well-ordering relations and induction. Correspondence between well-ordering relations and choice functions. (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.


Sample textbooks:

[1] Gordon Pace, Mathematics of Discrete Structures foe Computer Science, Springer, 2012

[2] Vladlen Koltun, Discrete Structures Lecture Notes, Stanford University, 2008. Freely available here.




Topics List

Week Topic Sections from Pace's book Sections from Pace's book Prerequisites.
1 Propositional logic 2.1-2.4
  • Proofs
  • boolean models
  • connections between boolean models and proofs
 MAT1313 or CS2233/2231, or equivalent.
2 Completeness and soundness 2.5
  • Completeness and soundness of propositional logic
5-6 Predicate calculus 3.1-3.5
  • Limits of propositional logic
  • free variables and substitution.
7 Sets and boolean algebras 4.1-4.5
  • Set comprehension.
  • Finitary and general operations on sets.
8 Sets and boolean algebras 4.6
  • Boolean algebras and boolean rings.
9 Relations 5.1-5.7
  • Relations and sets
  • Inverse of a relation and composition of relations
  • Beyond binary relations
10 Classifying Relations 6.1-6.3
  • Totality
  • Surjectivity
  • Injectivity
  • Functionality
11-12 Discrete structures 7.1-8.4
  • Graphs
  • Semigroups
  • groups
13-14 Reasoning about programs 10.1-10.4
  • Algorithms
  • Program semantics
  • Uncomputability
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