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Mathematics for Informatics
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[146289] Mathematics for Informatics - CHITTARO

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Mathematics for Computer Science
MIT 6.042J Mathematics for Computer Science, Spring 2015
Material and Exam mode
Fondamenti Matematici per l'Informatica
Peano's Axioms
Construction of Number Systems - N. Mohan Kumar
Recurrence Theorem
Chapter Four: The Natural Numbers, Induction, and Recursive Definition
The shifted induction principle
The Principle of Induction

Reference¶

Lecture 1
Sets
- 4.1 Sets, 4 Mathematical Data Types
Axiom of Extensionality
- 4.1.5 Proving Set Equalities
- 8.3.2 The ZFC Axioms for Sets
Subsets
- 4.1.2 Comparing and Combining Sets
- 8.3.2 The ZFC Axioms for Sets
- 10.7 Representing Partial Orders by Set Containment
Empty set
- 4.1.1 Some Popular Sets
Axiom of Specification
- 8.3.2 The ZFC Axioms for Sets
Operations between sets
- 4.1.2 Comparing and Combining Sets
Binary relations
- 4.4 Binary Relations
Functions
- 4.3 Functions
Injective functions, surjective functions, bijective functions
- 4.4.1 Relation Diagrams
Lecture 2
Identity function
- Problem 4.34.
Inverse of a bijective function
- 4.4.2 Relational Images
Inverse image of a set with respect to a function
- 4.4.2 Relational Images
Axiomatic definition of the set of natural numbers: Peano's axioms
- Construction of Number Systems - N. Mohan Kumar
Every non-zero natural number is the successor of a natural number
- Construction of Number Systems - N. Mohan Kumar
Lecture 3
Induction principle
- 5.1.1 A Rule for Ordinary Induction
Recurrence Theorem
- V Recurrences
Definition of the sum in the set N (using Recurrence theorem)
- Chapter Four: The Natural Numbers, Induction, and Recursive Definition
Lecture 4
Partial and total orderings
- 10.6 Partial Orders
The set of natural numbers is a totally ordered set
- Wikipedia
The shifted induction principle
- The Principle of Induction
Lecture 5
Well orderings
- 2 The Well Ordering Principle
The set of natural numbers is well ordered
- 2 The Well Ordering Principle
The strong induction principle
- 5.2.1 A Rule for Strong Induction
Lecture 6
Integer division
- 9.1 Divisibility
The Euclidean algorithm of division
- 9.2.1 Euclid’s Algorithm
Representability of integers in base b
- Positional notation
Greatest common divisor
- 9.2 The Greatest Common Divisor
Lecture 7
Greatest common divisor (existence and uniqueness)
- 9.2.4 Properties of the Greatest Common Divisor
Extended Euclidean Division Algorithm
- 9.2.2 The Pulverizer
Properties of coprime numbers
- 9.3 Prime Mysteries
Characterization of prime numbers
- 9.3 Prime Mysteries
Lecture 8
Least common multiple: definition, existence and uniqueness
- 9.4 The Fundamental Theorem of Arithmetic
The fundamental Theorem of Arithmetics
- 9.4 The Fundamental Theorem of Arithmetic
Prime numbers are infinite
- 9.4 The Fundamental Theorem of Arithmetic
Equivalence relations (definition)
- 10.10 Equivalence Relations
Lecture 9
Equivalence classes
- 10.10.1 Equivalence Classes
Congruences modulo n
- 9.6 Modular Arithmetic
The remainder classes
- 9.7 Remainder Arithmetic
The set Z/nZ
- 9.7.1 The ring Z_n
Addition and product of congruence classes
- 9.7.1 The ring Z_n
Lecture 10
The Chinese Remainder Theorem
- Problem 9.66.
Systems of Congruences
- 9.6 Modular Arithmetic
Lecture 11
Invertibility modulo n
- 9.9 Multiplicative Inverses and Cancelling
Cancellability modulo n
- 9.9.2 Cancellation
Linear equations mod n
- 10.10 Equivalence Relations