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Moodle Link
Textbook
About
Reference
- Lecture 1
- Sets
- Axiom of Extensionality
- Subsets
- Empty set
- Axiom of Specification
- Operations between sets
- Binary relations
- Functions
- Injective functions, surjective functions, bijective functions
- Lecture 2
- Identity function
- Inverse of a bijective function
- Inverse image of a set with respect to a function
- Axiomatic definition of the set of natural numbers: Peano's axioms
- Every non-zero natural number is the successor of a natural number
- Lecture 3
- Induction principle
- Recurrence Theorem
- Definition of the sum in the set N (using Recurrence theorem)
- Lecture 4
- Partial and total orderings
- The set of natural numbers is a totally ordered set
- The shifted induction principle
- Lecture 5
- Well orderings
- The set of natural numbers is well ordered
- The strong induction principle
- Lecture 6
- Integer division
- The Euclidean algorithm of division
- Representability of integers in base b
- Greatest common divisor
- Lecture 7
- Greatest common divisor (existence and uniqueness)
- Extended Euclidean Division Algorithm
- Properties of coprime numbers
- Characterization of prime numbers
- Lecture 8
- Least common multiple: definition, existence and uniqueness
- The fundamental Theorem of Arithmetics
- Prime numbers are infinite
- Equivalence relations (definition)
- Lecture 9
- Equivalence classes
- Congruences modulo n
- The remainder classes
- The set Z/nZ
- Addition and product of congruence classes
- Lecture 10
- The Chinese Remainder Theorem
- Systems of Congruences
- Lecture 11
- Invertibility modulo n
- Cancellability modulo n
- Linear equations mod n