Difference between revisions of "MATxxx"

From Department of Mathematics at UTSA
Jump to navigation Jump to search
(Created page with "1. Propositional logic: Axioms and Rules of Inference. Boolean Algebras. Limitations of propositional logic: Informal introduction to quantifiers and syllogisms. 2. Predicate...")
 
Line 1: Line 1:
−
1. Propositional logic: Axioms and Rules of Inference. Boolean Algebras. Limitations of propositional logic: Informal introduction to quantifiers and syllogisms.
+
(1) Relations: Cartesian products, relations, properties of relations, equivalence relations and partitions. (2) Order relations: Partially ordered sets, totally ordered sets, extreme elements (maximum, minimum, maximal and minimal elements), well-ordered sets, maximality principles, Zorn's Lemma, lattices, boolean algebras, circuit design. (3) Graphs: Euler and Hamiltonian paths and circuits, matching, graph coloring, Ramsey’s theorem, trees and searching. (4) Binary operations: Groups and semigroups, products and quotients of groups, other algebraic structures.
−
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: 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: Digital vs analog vs quantum computing.
 

Revision as of 14:30, 9 March 2023

(1) Relations: Cartesian products, relations, properties of relations, equivalence relations and partitions. (2) Order relations: Partially ordered sets, totally ordered sets, extreme elements (maximum, minimum, maximal and minimal elements), well-ordered sets, maximality principles, Zorn's Lemma, lattices, boolean algebras, circuit design. (3) Graphs: Euler and Hamiltonian paths and circuits, matching, graph coloring, Ramsey’s theorem, trees and searching. (4) Binary operations: Groups and semigroups, products and quotients of groups, other algebraic structures.

Retrieved from "https://mathresearch.utsa.edu/wiki/index.php?title=MATxxx&oldid=4741"