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2023-03-09T21:05:26Z

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Jose.iovino at 20:30, 9 March 2023

2023-03-09T20:30:34Z

Jose.iovino: 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..."

2023-03-09T20:24:04Z

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..."

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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: 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.

← Older revision		Revision as of 21:05, 9 March 2023
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−	(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.	+

← Older revision		Revision as of 20:30, 9 March 2023
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~~.