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| − | Introduction to basic discrete structures.
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| − | '''Sample textbooks''':
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| − | [1] Gordon Pace, ''Mathematics of Discrete Structures foe Computer Science'', Springer, 2012
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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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| | '''Catalog entry''' | | '''Catalog entry''' |
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| − | ''Prerequisite'': Algebra and Number Systems (MAT 1313), or Discrete Mathematical Structures (CS 2233/2231), or instructor consent. | + | ''Prerequisite'': Combinatorics and Probability [[MAT2313]], or Applied Graph Theory [[MAT4323]], or instructor consent. |
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| | ''Contents'': | | ''Contents'': |
| − | (1) Propositional logic: Axioms and Rules of Inference. Boolean Algebras. Limitations of propositional logic: Informal introduction to quantifiers and syllogisms.
| + | Partially ordered sets, maximum/maximal and minimum/minimal elements. Well-ordered sets. Maximality principlies (Zorn's lemma, Well-ordering principle, Hausdorff maximality lemma). Boolean algebras and the Stone representation theorem. Generalizations of the Stone representation theorem. |
| − | (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: Properties of relations. Special relations: Equivalence relations, partially ordered sets, totally ordered sets.
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| − | (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.
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| − | (7) Introduction to computability. Classical models of computation (recursive functions, and Turing models). Limitations of computation (the Halting Problem, the busy beaver problem, fast-growing functions). Contemporary models of computation.
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| − | ==Topics List==
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| − | {| class="wikitable sortable"
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| − | ! Week !! Topic !! Sections from Pace's book !! Calculus 2
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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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| − | || Undergraduate real analysis.
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| − | |-
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| − | | 3-4
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| − | || [[Completeness and soundness]]
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| − | || 2.5-2.7.
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| − | ||
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| − | |-
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| − | | 4-5
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| − | || [[Predicate calculus]]
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| − | || 3.1-3.5
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| − | ||
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| − | |-
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| − | | 6-7
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| − | || [[Sets and boolean algebras]]
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| − | || 4.1-4.8
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| − | ||
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| − | |-
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| − | | 8
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| − | || [[Relations]]
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| − | || 5.1-6.3
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| − | ||
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| − | |-
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| − | | 9-10
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| − | || [[Discrete structures]]
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| − | || 7.1-8.4
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| − | ||
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| − | |-
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| − | | 10-16
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| − | || [[Models of computation]]
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| − | || 10.1-10.4
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| − | ||
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| − | |}
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Catalog entry
Prerequisite: Combinatorics and Probability MAT2313, or Applied Graph Theory MAT4323, or instructor consent.
Contents:
Partially ordered sets, maximum/maximal and minimum/minimal elements. Well-ordered sets. Maximality principlies (Zorn's lemma, Well-ordering principle, Hausdorff maximality lemma). Boolean algebras and the Stone representation theorem. Generalizations of the Stone representation theorem.
