MATH 347H Fall 2020

MATH 347H Fall 2020

Show Details

Lecture 5.2 Part 5: Archimedean property

duration 8 minutes 58 seconds

Lecture 5.2 Part 5: Archimedean property

Lecture 5.2 Part 5: Archimedean property

From Ruiyuan Chen

36 plays 0
The Archimedean property, density of Q, every real number is a supremum of rationals
Lecture 5.2 Part 4: Dedekind-completeness

duration 10 minutes 37 seconds

Lecture 5.2 Part 4: Dedekind-completeness

Lecture 5.2 Part 4: Dedekind-completeness

From Ruiyuan Chen

24 plays 0
Dedekind-completeness, equivalence with dual condition for infima, characterization of R as unique-up-to-unique-isomorphism Dedekind-complete ordered field
Lecture 5.2 Part 3: Suprema and infima

duration 10 minutes 6 seconds

Lecture 5.2 Part 3: Suprema and infima

Lecture 5.2 Part 3: Suprema and infima

From Ruiyuan Chen

27 plays 0
Upper and lower bounds, suprema and infima, relationship to max and min
Lecture 5.2 Part 2: Ordered fields

duration 14 minutes 41 seconds

Lecture 5.2 Part 2: Ordered fields

Lecture 5.2 Part 2: Ordered fields

From Ruiyuan Chen

30 plays 0
Ordered rings and fields, basic consequences of axioms, Q is the smallest ordered field up to isomorphism
Lecture 5.2 Part 1: Motivation for real numbers

duration 6 minutes 8 seconds

Lecture 5.2 Part 1: Motivation for real numbers

Lecture 5.2 Part 1: Motivation for real numbers

From Ruiyuan Chen

27 plays 0
Lack of square roots in Q, preview of connection between square roots and limits & definition of limits via ordering
Lecture 5.1 Part 4: Construction of Q

duration 10 minutes 19 seconds

Lecture 5.1 Part 4: Construction of Q

Lecture 5.1 Part 4: Construction of Q

From Ruiyuan Chen

25 plays 0
Adjoining inverses to general commutative monoids, construction of Q from Z
Lecture 5.1 Part 3: Other structure on Z

duration 9 minutes 3 seconds

Lecture 5.1 Part 3: Other structure on Z

Lecture 5.1 Part 3: Other structure on Z

From Ruiyuan Chen

25 plays 0
The embedding of N into Z, cancellation of multiplication in Z, ordering in Z
Lecture 5.1 Part 2: Arithmetic on Z

duration 11 minutes 40 seconds

Lecture 5.1 Part 2: Arithmetic on Z

Lecture 5.1 Part 2: Arithmetic on Z

From Ruiyuan Chen

26 plays 0
Definition of addition and multiplication on Z, proof of commutative ring axioms
Lecture 5.1 Part 1: Construction of Z

duration 6 minutes 28 seconds

Lecture 5.1 Part 1: Construction of Z

Lecture 5.1 Part 1: Construction of Z

From Ruiyuan Chen

53 plays 0
Construction of Z from N
Lecture 4.6 Part 3: Descent

duration 20 minutes 24 seconds

Lecture 4.6 Part 3: Descent

Lecture 4.6 Part 3: Descent

From Ruiyuan Chen

39 plays 0
Descent of functions to quotient sets, descent of multi-variable functions, quotient structures and congruence relations
Lecture 4.6 Part 2: Partitions & surjections

duration 12 minutes 6 seconds

Lecture 4.6 Part 2: Partitions & surjections

Lecture 4.6 Part 2: Partitions & surjections

From Ruiyuan Chen

27 plays 0
Correspondence between equivalence relations, partitions, and surjections; integers mod d
Lecture 4.6 Part 1: Equivalence relations

duration 13 minutes 18 seconds

Lecture 4.6 Part 1: Equivalence relations

Lecture 4.6 Part 1: Equivalence relations

From Ruiyuan Chen

38 plays 0
Equivalence relations, congruence mod d, kernels, cardinality, quotient sets
Lecture 4.5 Part 2: Relational substructures…

duration 12 minutes 31 seconds

Lecture 4.5 Part 2: Relational substructures…

Lecture 4.5 Part 2: Relational substructures & morphisms

From Ruiyuan Chen

28 plays 0
Relational substructures vs induced substructures, homomorphisms, bijective homomorphisms vs isomorphisms
Lecture 4.5 Part 1: Relational structures

duration 14 minutes 9 seconds

Lecture 4.5 Part 1: Relational structures

Lecture 4.5 Part 1: Relational structures

From Ruiyuan Chen

31 plays 0
Digraphs, reflexivity, symmetry, transitivity, antisymmetry, trichotomy, preorders, partial orders, total orders, equivalence relations
Lecture 4.4 Part 4: More isomorphisms

duration 11 minutes 32 seconds

Lecture 4.4 Part 4: More isomorphisms

Lecture 4.4 Part 4: More isomorphisms

From Ruiyuan Chen

22 plays 0
The fundamental theorem of arithmetic as a monoid isomorphism, transport of properties and constructions along isomorphisms
Lecture 4.4 Part 3: Isomorphisms

duration 11 minutes 45 seconds

Lecture 4.4 Part 3: Isomorphisms

Lecture 4.4 Part 3: Isomorphisms

From Ruiyuan Chen

27 plays 0
Isomorphisms, vs bijective homomorphisms, exponential isomorphism between addition and multiplication
Lecture 4.4 Part 2: More homomorphisms

duration 7 minutes 19 seconds

Lecture 4.4 Part 2: More homomorphisms

Lecture 4.4 Part 2: More homomorphisms

From Ruiyuan Chen

27 plays 0
Image and preimage as monoid homomorphisms, image and preimage of substructures under homomorphisms
Lecture 4.4 Part 1: Homomorphisms

duration 13 minutes 57 seconds

Lecture 4.4 Part 1: Homomorphisms

Lecture 4.4 Part 1: Homomorphisms

From Ruiyuan Chen

30 plays 0
Homomorphisms, semigroup vs group homomorphisms, numeric examples
Lecture 4.3 Part 2: Fields

duration 9 minutes 31 seconds

Lecture 4.3 Part 2: Fields

Lecture 4.3 Part 2: Fields

From Ruiyuan Chen

27 plays 0
Inverses in rings, division rings, fields
Lecture 4.3 Part 1: Rings

duration 17 minutes 11 seconds

Lecture 4.3 Part 1: Rings

Lecture 4.3 Part 1: Rings

From Ruiyuan Chen

33 plays 0
Rings, subrings, basic consequences of ring axioms, trivial rings, function rings
Lecture 4.2 Part 2: Subgroups

duration 19 minutes 27 seconds

Lecture 4.2 Part 2: Subgroups

Lecture 4.2 Part 2: Subgroups

From Ruiyuan Chen

31 plays 0
Subgroups, intersections and generated subgroups, powers of elements in groups, generated subgroups in Z and gcd's
Lecture 4.2 Part 1: Inverses and groups

duration 13 minutes 57 seconds

Lecture 4.2 Part 1: Inverses and groups

Lecture 4.2 Part 1: Inverses and groups

From Ruiyuan Chen

38 plays 0
Left and right inverses in monoids, groups, group of invertible elements in a monoid
Lecture 4.1 Part 3: Submonoids

duration 19 minutes 29 seconds

Lecture 4.1 Part 3: Submonoids

Lecture 4.1 Part 3: Submonoids

From Ruiyuan Chen

37 plays 0
Submonoids, intersection of submonoids, submonoid generated by a subset, submonoids and equational axioms
Lecture 4.1 Part 2: Monoids

duration 12 minutes 29 seconds

Lecture 4.1 Part 2: Monoids

Lecture 4.1 Part 2: Monoids

From Ruiyuan Chen

41 plays 0
(Commutative) monoids, numeric and set examples, functions under composition, free monoids
Lecture 4.1 Part 1: Abstract structures

duration 4 minutes 54 seconds

Lecture 4.1 Part 1: Abstract structures

Lecture 4.1 Part 1: Abstract structures

From Ruiyuan Chen

40 plays 0
General philosophy of abstract structures
Lecture 3.5 Part 3: Prime factorization

duration 7 minutes 53 seconds

Lecture 3.5 Part 3: Prime factorization

Lecture 3.5 Part 3: Prime factorization

From Ruiyuan Chen

20 plays 0
Euclid's lemma, the fundamental theorem of arithmetic
Lecture 3.5 Part 2: Division and gcd's

duration 16 minutes 25 seconds

Lecture 3.5 Part 2: Division and gcd's

Lecture 3.5 Part 2: Division and gcd's

From Ruiyuan Chen

28 plays 0
Euclidean division with remainder, greatest common divisors, coprime integers, the Euclidean algorithm and Bézout's identity
Lecture 3.5 Part 1: Divisibility and primes

duration 10 minutes 51 seconds

Lecture 3.5 Part 1: Divisibility and primes

Lecture 3.5 Part 1: Divisibility and primes

From Ruiyuan Chen

28 plays 0
Divisibility, primes, infinitude of primes
Lecture 3.4: Binomial coefficients

duration 19 minutes 13 seconds

Lecture 3.4: Binomial coefficients

Lecture 3.4: Binomial coefficients

From Ruiyuan Chen

35 plays 0
Binomial coefficients, symmetry, Pascal's triangle, Pascal's identity, diagonal sums of Pascal's triangle, the binomial theorem, row sums of Pascal's…
Lecture 3.3: Induction as an axiom

duration 14 minutes 52 seconds

Lecture 3.3: Induction as an axiom

Lecture 3.3: Induction as an axiom

From Ruiyuan Chen

35 plays 0
Set-theoretic definition of N and successor, the axiom of infinity, definition of arithmetic, preview of construction of Z, Q, R

Search for ""

Public, Restricted

66 Media

1 Members

Managers:: Ruiyuan Chen