MATH 347H Fall 2020
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From Ruiyuan Chen 12/16/2020
38 plays 0The Archimedean property, density of Q, every real number is a supremum of rationals -
From Ruiyuan Chen 12/15/2020
25 plays 0Dedekind-completeness, equivalence with dual condition for infima, characterization of R as unique-up-to-unique-isomorphism Dedekind-complete ordered field -
From Ruiyuan Chen 12/11/2020
27 plays 0Upper and lower bounds, suprema and infima, relationship to max and min -
From Ruiyuan Chen 12/11/2020
30 plays 0Ordered rings and fields, basic consequences of axioms, Q is the smallest ordered field up to isomorphism -
From Ruiyuan Chen 12/10/2020
27 plays 0Lack of square roots in Q, preview of connection between square roots and limits & definition of limits via ordering -
From Ruiyuan Chen 12/9/2020
25 plays 0Adjoining inverses to general commutative monoids, construction of Q from Z -
From Ruiyuan Chen 12/9/2020
25 plays 0The embedding of N into Z, cancellation of multiplication in Z, ordering in Z -
From Ruiyuan Chen 12/5/2020
26 plays 0Definition of addition and multiplication on Z, proof of commutative ring axioms -
From Ruiyuan Chen 12/5/2020
56 plays 0Construction of Z from N -
From Ruiyuan Chen 11/19/2020
39 plays 0Descent of functions to quotient sets, descent of multi-variable functions, quotient structures and congruence relations -
From Ruiyuan Chen 11/19/2020
27 plays 0Correspondence between equivalence relations, partitions, and surjections; integers mod d -
From Ruiyuan Chen 11/18/2020
38 plays 0Equivalence relations, congruence mod d, kernels, cardinality, quotient sets -
Lecture 4.5 Part 2: Relational substructures… 13 of 30
Lecture 4.5 Part 2: Relational substructures…
From Ruiyuan Chen 11/17/2020
28 plays 0Relational substructures vs induced substructures, homomorphisms, bijective homomorphisms vs isomorphisms -
From Ruiyuan Chen 11/16/2020
31 plays 0Digraphs, reflexivity, symmetry, transitivity, antisymmetry, trichotomy, preorders, partial orders, total orders, equivalence relations -
From Ruiyuan Chen 11/12/2020
22 plays 0The fundamental theorem of arithmetic as a monoid isomorphism, transport of properties and constructions along isomorphisms -
From Ruiyuan Chen 11/12/2020
27 plays 0Isomorphisms, vs bijective homomorphisms, exponential isomorphism between addition and multiplication -
From Ruiyuan Chen 11/11/2020
27 plays 0Image and preimage as monoid homomorphisms, image and preimage of substructures under homomorphisms -
From Ruiyuan Chen 11/11/2020
30 plays 0Homomorphisms, semigroup vs group homomorphisms, numeric examples -
From Ruiyuan Chen 11/8/2020
27 plays 0Inverses in rings, division rings, fields -
From Ruiyuan Chen 11/8/2020
33 plays 0Rings, subrings, basic consequences of ring axioms, trivial rings, function rings -
From Ruiyuan Chen 11/4/2020
31 plays 0Subgroups, intersections and generated subgroups, powers of elements in groups, generated subgroups in Z and gcd's -
From Ruiyuan Chen 11/3/2020
38 plays 0Left and right inverses in monoids, groups, group of invertible elements in a monoid -
From Ruiyuan Chen 10/30/2020
37 plays 0Submonoids, intersection of submonoids, submonoid generated by a subset, submonoids and equational axioms -
From Ruiyuan Chen 10/28/2020
41 plays 0(Commutative) monoids, numeric and set examples, functions under composition, free monoids -
From Ruiyuan Chen 10/28/2020
40 plays 0General philosophy of abstract structures -
From Ruiyuan Chen 10/20/2020
20 plays 0Euclid's lemma, the fundamental theorem of arithmetic -
From Ruiyuan Chen 10/20/2020
28 plays 0Euclidean division with remainder, greatest common divisors, coprime integers, the Euclidean algorithm and Bézout's identity -
From Ruiyuan Chen 10/20/2020
28 plays 0Divisibility, primes, infinitude of primes -
From Ruiyuan Chen 10/19/2020
35 plays 0Binomial coefficients, symmetry, Pascal's triangle, Pascal's identity, diagonal sums of Pascal's triangle, the binomial theorem, row sums of Pascal's… -
From Ruiyuan Chen 10/19/2020
38 plays 0Set-theoretic definition of N and successor, the axiom of infinity, definition of arithmetic, preview of construction of Z, Q, R