Difference between revisions of "MAT5423"

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(1) Propositional logic: Axioms and Rules of Inference. Boolean Algebras. Limitations of propositional logic: Informal introduction to quantifiers and syllogisms.
 
(1) Propositional logic: Axioms and Rules of Inference. Boolean Algebras. 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.
 
(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.
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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.
 
(4) Relations: Properties of relations. Special relations: Equivalence relations, partially ordered sets, totally ordered sets.
 
(4) Relations: Properties of relations. Special relations: Equivalence relations, partially ordered sets, totally ordered sets.
 
(5) Functions: Operations of functions, direct image and inverse image.
 
(5) Functions: Operations of functions, direct image and inverse image.

Revision as of 15:18, 24 March 2023

Introduction to basic discrete structures.

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.


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. Boolean Algebras. 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: Properties of relations. Special relations: Equivalence relations, partially ordered sets, 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, the busy beaver problem, fast-growing functions). Contemporary models of computation.




Topics List

Week Topic Sections from Pace's book MAT1313, or CS2233/2231, or instructor consent.
1 Propositional logic 2.1-2.4 MAT1313 or CS2233/2231, or equivalent.
2 Completeness and soundness 2.5-2.7.
3-4 Predicate calculus 3.1-3.5
5-6 Sets and boolean algebras 4.1-4.8
7-8 Relations 5.1-6.3
9-10 Discrete structures 7.1-8.4
11-13 Mathematical models of computation 10.1-10.4