In music theory, equivalence class is an equality (=) or equivalence between properties of sets (unordered) or twelve-tone rows (ordered sets). A relation rather than an operation, it may be contrasted with derivation. "It is not surprising that music theorists have different concepts of equivalence [from each other]..." "Indeed, an informal notion of equivalence has always been part of music theory and analysis. Pitch class set theory, however, has adhered to formal definitions of equivalence." Traditionally, octave equivalency is assumed, while inversional, permutational, and transpositional equivalency may or may not be considered (sequences and modulations are techniques of the common practice period which are based on transpositional equivalency; similarity within difference; unity within variety/variety within unity). A definition of equivalence between two twelve-tone series that Schuijer describes as informal despite its air of mathematical precision, and that shows its writer considered equivalence and equality as synonymous: Two sets [twelve-tone series], P and P′ will be considered equivalent [equal] if and only if, for any pi,j of the first set and p′i′,j′ of the second set, for all is and js [order numbers and pitch class numbers], if i=i′, then j=j′. (= denotes numeral equality in the ordinary sense). — Milton Babbitt, (1992). The Function of Set Structure in the Twelve-Tone System, 8-9, cited in Forte (1963, p. 76) similarly uses equivalent to mean identical, "considering two subsets as equivalent when they consisted of the same elements. In such a case, mathematical set theory speaks of the 'equality,' not the 'equivalence,' of sets." However, equality may be considered identical (equivalent in all ways) and thus contrasted with equivalence and similarity (equivalent in one or more ways but not all). For example, the C major scale, G major scale, and the major scale in all keys, are not identical but share transpositional equivalence in that the size of the intervals between scale steps is identical while pitches are not (C major has F♮ while G major has F♯). The major third and the minor sixth are not identical but share inversional equivalence (an inverted M3 is a m6, an inverted m6 is a M3). A melody with the notes G A B C is not identical to a melody with the notes C B A G, but they share retrograde equivalence. (Wikipedia).
Two Equivalence Classes [a] and [b] Are Equal If and Only If a is Related to b
In this video I prove a statement surrounding relations. We have an equivalence relation on a set A and we have to show that the equivalence class of a is equal to the equivalence class of b if and only if a is related to b. If you enjoyed this video please consider liking, sharing, and
From playlist Relations
Cosets and equivalence class proof
Now that we have shown that the relation on G is an equivalence relation ( https://www.youtube.com/watch?v=F7OgJi6o9po ), we can go on to prove that the equivalence class containing an element is the same as the corresponding set on H (a subset of G).
From playlist Abstract algebra
The picture in the lecture was taken from Wikipedia: https://en.wikipedia.org/wiki/Demographics_of_the_United_States#/media/File:USA2020dec1.png
From playlist Abstract Algebra 1
Important Math Proof: The Set of Equivalence Classes Partition a Set
In this video I prove a very important result in mathematics. Given an equivalence relation R on a nonempty set A, the set S of equivalence classes of A is a partition of A. Stated another way, this result says we can write A as a disjoint union of equivalence classes. The pencils I used
From playlist Relations
Discrete Math - 9.5.1 Equivalence Relations
Exploring a special kind of relation, called an equivalence relation. Equivalence classes and partitions are also discussed. Textbook: Rosen, Discrete Mathematics and Its Applications, 7e Playlist: https://www.youtube.com/playlist?list=PLl-gb0E4MII28GykmtuBXNUNoej-vY5Rz
From playlist Discrete Math I (Entire Course)
This video is a full introduction to equivalence relations. Timestamps: 0:00 What is a relation? 3:02 Terminology - A Relation defined on a Set 4:02 Equivalence Relation Definition 7:18 Reflexive 9:18 Symmetric 11:48 Transitive Thanks for watching! Comment below with questions, and make
From playlist Proofs
Put all three properties of binary relations together and you have an equivalence relation.
From playlist Abstract algebra
Equivalences and Partitions, Axiomatic Set Theory 2 2
Defining equivalences and partitions of sets, and proving some theorems about their relations to each other. My Twitter: https://twitter.com/KristapsBalodi3 Equivalence Relations:(0:00) Partitions:(9:22) Connecting Equivalence and Partitions:(14:09) Representatives:(27:04)
From playlist Axiomatic Set Theory
Set Theory (Part 6): Equivalence Relations and Classes
Please feel free to leave comments/questions on the video and practice problems below! In this video, I set up equivalence relations and the canonical mapping. The idea of equivalence relation will return when we construct higher-level number systems, e.g.integers, from the natural number
From playlist Set Theory by Mathoma
Intro to proofs April 20, 2021
Suggest a problem: https://forms.gle/ea7Pw7HcKePGB4my5 Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Merch: https://teespring.com/stores/michael-penn-math Personal Website: http://www.michael-penn.net Randolph College Math: http://www.randolphcollege.edu/ma
From playlist Super Lo-fi in class videos
Henrique Bursztyn: Relating Morita equivalence in algebra and geometry via deformation quantization
Talk by Henrique Bursztyn in Global Noncommutative Geometry Seminar (Americas) https://globalncgseminar.org/talks/3225/ on April 2, 2021.
From playlist Global Noncommutative Geometry Seminar (Americas)
Ralf Meyer: On the classification of group actions on C*-algebras up to equivariant KK-equivalence
Talk by Ralf Meyer in Global Noncommutative Geometry Seminar (Europe) http://www.noncommutativegeometry.nl/ncgseminar/ on November 10, 2020.
From playlist Global Noncommutative Geometry Seminar (Europe)
Basic Homotopy Theory by Samik Basu
PROGRAM DUALITIES IN TOPOLOGY AND ALGEBRA (ONLINE) ORGANIZERS: Samik Basu (ISI Kolkata, India), Anita Naolekar (ISI Bangalore, India) and Rekha Santhanam (IIT Mumbai, India) DATE & TIME: 01 February 2021 to 13 February 2021 VENUE: Online Duality phenomena are ubiquitous in mathematics
From playlist Dualities in Topology and Algebra (Online)
Stable Homotopy Seminar, 4: Model categories (Ivo Vekemans)
This talk by Ivo Vekemans is a thorough introduction to model categories, presenting: weak factorization systems; the definition of model category and major examples (simplicial sets, topological spaces, and chain complexes); notions of homotopy in a model category, and the homotopy catego
From playlist Stable Homotopy Seminar
Suggest a problem: https://forms.gle/ea7Pw7HcKePGB4my5 Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Merch: https://teespring.com/stores/michael-penn-math Personal Website: http://www.michael-penn.net Randolph College Math: http://www.randolphcollege.edu/ma
From playlist Super Lo-fi in class videos
Stefan Schwede: Equivariant stable homotopy - Lecture 2
I will use the orthogonal spectrum model to introduce the tensor triangulated category of genuine G-spectra, for compact Lie groups G. I will explain structural properties such as the smash product of G-spectra, and functors relating the categories for varying G (fixed points, geometric fi
From playlist Summer School: Spectral methods in algebra, geometry, and topology
Lecture 15. Kernels and cokernels
From playlist Abstract Algebra 2
Volker Genz - Reddening sequences for Cluster Algebras
While cluster algebras generally are not finitely generated, reddening sequences offer a more relaxed notion of finiteness. The existence of a redden- ing sequence has far reaching consequences for a cluster algebra (generic finite dimensionality of the Jacobian, numeric Donaldson-Thomas i
From playlist Combinatorics and Arithmetic for Physics: Special Days 2022
Equivalence Relations Definition and Examples
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Equivalence Relations Definition and Examples. This video starts by defining a relation, reflexive relation, symmetric relation, transitive relation, and then an equivalence relation. Several examples are given.
From playlist Abstract Algebra
François Métayer: Homotopy theory of strict omega-categories and its connections with...Part 1
Abstract: In the first part, we describe the canonical model structure on the category of strict ω-categories and how it transfers to related subcategories. We then characterize the cofibrant objects as ω-categories freely generated by polygraphs and introduce the key notion of polygraphic
From playlist Topology