Lecture

Equivalence Relations

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

In course

MATH-113: Algebra I - fundamental structures

Le but de ce cours est d'introduire et d'étudier les notions de base de l'algèbre abstraite. ...

Description

This lecture covers the concept of equivalence relations, including reflexivity, symmetry, and transitivity. It explains how to determine if a relation is an equivalence relation and provides examples to illustrate the concept.

Instructors (2)

Dimitri Stelio Wyss

I did my PhD under the supervision of Tamas Hausel at EPFL and the Institue of Science and Technology Austria . From 2017-2019 I was a postdoctoral researcher at the Sorbonne University Paris 6 with François Loeser. Since September 2019 I hold the chair of arithmetic geometry at EPFL as a tenure track assistant professor. For more information see my CV.I work in algebraic and arithmetic geometry. My research uses ideas from p-adic and motivic integration to study the interplay between geometric and arithmet

Zsolt Patakfalvi

See CV here .

Official source

Related concepts (19)

Symmetry

Symmetry () in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations, such as translation, reflection, rotation, or scaling. Although these two meanings of the word can sometimes be told apart, they are intricately related, and hence are discussed together in this article.

Symmetry group

In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object X is G = Sym(X). For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space.

Rotational symmetry

Rotational symmetry, also known as radial symmetry in geometry, is the property a shape has when it looks the same after some rotation by a partial turn. An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation. Certain geometric objects are partially symmetrical when rotated at certain angles such as squares rotated 90°, however the only geometric objects that are fully rotationally symmetric at any angle are spheres, circles and other spheroids.

Icosahedral symmetry

In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the icosahedron) and the rhombic triacontahedron. Every polyhedron with icosahedral symmetry has 60 rotational (or orientation-preserving) symmetries and 60 orientation-reversing symmetries (that combine a rotation and a reflection), for a total symmetry order of 120.

Reflection symmetry

In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D there is a line/axis of symmetry, in 3D a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called . In conclusion, a line of symmetry splits the shape in half and those halves should be identical.

Related lectures (64)

Equivalence Relations: Classes, Partitions

Explains equivalence relations, classes, and partitions in sets.

Equivalence Relations and Partitions

Discusses equivalence relations, partitions, reflexivity, symmetry, transitivity, and set cardinality.

Inertia Tensor: Main Axes and Principal Moments

Explains the inertia tensor, main axes, principal moments, and balancing of rotating solids.

Relations and Sequences

Covers relations, sequences, and posets, emphasizing properties like anti-symmetry and transitivity, and introduces arithmetic and geometric progressions.

Global Conservation Laws