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The Archimedean property, density of Q, every real number is a supremum of rationals
Lecture 5.2 Part 5: Archimedean property
From Ruiyuan Chen
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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 4: Dedekind-completeness
From Ruiyuan Chen
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Upper and lower bounds, suprema and infima, relationship to max and min
Lecture 5.2 Part 3: Suprema and infima
From Ruiyuan Chen
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Ordered rings and fields, basic consequences of axioms, Q is the smallest ordered field up to isomorphism
Lecture 5.2 Part 2: Ordered fields
From Ruiyuan Chen
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Lack of square roots in Q, preview of connection between square roots and limits & definition of limits via ordering
Lecture 5.2 Part 1: Motivation for real numbers
From Ruiyuan Chen
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Adjoining inverses to general commutative monoids, construction of Q from Z
Lecture 5.1 Part 4: Construction of Q
From Ruiyuan Chen
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The embedding of N into Z, cancellation of multiplication in Z, ordering in Z
Lecture 5.1 Part 3: Other structure on Z
From Ruiyuan Chen
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Definition of addition and multiplication on Z, proof of commutative ring axioms
Lecture 5.1 Part 2: Arithmetic on Z
From Ruiyuan Chen
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Construction of Z from N
Lecture 5.1 Part 1: Construction of Z
From Ruiyuan Chen
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Descent of functions to quotient sets, descent of multi-variable functions, quotient structures and congruence relations
Lecture 4.6 Part 3: Descent
From Ruiyuan Chen
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Correspondence between equivalence relations, partitions, and surjections; integers mod d
Lecture 4.6 Part 2: Partitions & surjections
From Ruiyuan Chen
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Equivalence relations, congruence mod d, kernels, cardinality, quotient sets
Lecture 4.6 Part 1: Equivalence relations
From Ruiyuan Chen
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Lecture 4.5 Part 2: Relational substructures… 13 of 30
Lecture 4.5 Part 2: Relational substructures…
Relational substructures vs induced substructures, homomorphisms, bijective homomorphisms vs isomorphismsLecture 4.5 Part 2: Relational substructures…
From Ruiyuan Chen
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Digraphs, reflexivity, symmetry, transitivity, antisymmetry, trichotomy, preorders, partial orders, total orders, equivalence relations
Lecture 4.5 Part 1: Relational structures
From Ruiyuan Chen
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The fundamental theorem of arithmetic as a monoid isomorphism, transport of properties and constructions along isomorphisms
Lecture 4.4 Part 4: More isomorphisms
From Ruiyuan Chen
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Isomorphisms, vs bijective homomorphisms, exponential isomorphism between addition and multiplication
Lecture 4.4 Part 3: Isomorphisms
From Ruiyuan Chen
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Image and preimage as monoid homomorphisms, image and preimage of substructures under homomorphisms
Lecture 4.4 Part 2: More homomorphisms
From Ruiyuan Chen
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Homomorphisms, semigroup vs group homomorphisms, numeric examples
Lecture 4.4 Part 1: Homomorphisms
From Ruiyuan Chen
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Inverses in rings, division rings, fields
Lecture 4.3 Part 2: Fields
From Ruiyuan Chen
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Rings, subrings, basic consequences of ring axioms, trivial rings, function rings
Lecture 4.3 Part 1: Rings
From Ruiyuan Chen
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Subgroups, intersections and generated subgroups, powers of elements in groups, generated subgroups in Z and gcd's
Lecture 4.2 Part 2: Subgroups
From Ruiyuan Chen
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Left and right inverses in monoids, groups, group of invertible elements in a monoid
Lecture 4.2 Part 1: Inverses and groups
From Ruiyuan Chen
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Submonoids, intersection of submonoids, submonoid generated by a subset, submonoids and equational axioms
Lecture 4.1 Part 3: Submonoids
From Ruiyuan Chen
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(Commutative) monoids, numeric and set examples, functions under composition, free monoids
Lecture 4.1 Part 2: Monoids
From Ruiyuan Chen
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General philosophy of abstract structures
Lecture 4.1 Part 1: Abstract structures
From Ruiyuan Chen
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Euclid's lemma, the fundamental theorem of arithmetic
Lecture 3.5 Part 3: Prime factorization
From Ruiyuan Chen
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Euclidean division with remainder, greatest common divisors, coprime integers, the Euclidean algorithm and Bézout's identity
Lecture 3.5 Part 2: Division and gcd's
From Ruiyuan Chen
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Divisibility, primes, infinitude of primes
Lecture 3.5 Part 1: Divisibility and primes
From Ruiyuan Chen
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Binomial coefficients, symmetry, Pascal's triangle, Pascal's identity, diagonal sums of Pascal's triangle, the binomial theorem, row sums of Pascal's…
Lecture 3.4: Binomial coefficients
From Ruiyuan Chen
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Set-theoretic definition of N and successor, the axiom of infinity, definition of arithmetic, preview of construction of Z, Q, R
Lecture 3.3: Induction as an axiom
From Ruiyuan Chen
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