Concept

Set (mathematics)

A set is the mathematical model for a collection of different things; a set contains elements or members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an infinite set. Two sets are equal if they have precisely the same elements. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. Set theory The concept of a set emerged in mathematics at the end of the 19th century. The German word for set, Menge, was coined by Bernard Bolzano in his work Paradoxes of the Infinite. Georg Cantor, one of the founders of set theory, gave the following definition at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre: A set is a gathering together into a whole of definite, distinct objects of our perception or our thought—which are called elements of the set. Bertrand Russell introduced the distinction between a set and a class (a set is a class, but some classes, such as the class of all sets, are not sets; see Russell's paradox): When mathematicians deal with what they call a manifold, aggregate, Menge, ensemble, or some equivalent name, it is common, especially where the number of terms involved is finite, to regard the object in question (which is in fact a class) as defined by the enumeration of its terms, and as consisting possibly of a single term, which in that case is the class. Naive set theory The foremost property of a set is that it can have elements, also called members. Two sets are equal when they have the same elements. More precisely, sets A and B are equal if every element of A is an element of B, and every element of B is an element of A; this property is called the extensionality of sets.

Official source

https://en.wikipedia.org/wiki/Set_(mathematics)

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related courses (32)

MATH-410: Riemann surfaces

This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex domains under discontinuous group actions, as algebraic ...

COM-406: Foundations of Data Science

We discuss a set of topics that are important for the understanding of modern data science but that are typically not taught in an introductory ML course. In particular we discuss fundamental ideas and techniques that come from probability, information the ...

PHYS-467: Machine learning for physicists

Machine learning and data analysis are becoming increasingly central in sciences including physics. In this course, fundamental principles and methods of machine learning will be introduced and practised. ...

Related units (32)

L'IDIAP Laboratory

Transportation and Mobility Laboratory

Polymers Laboratory

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.

Official source

https://en.wikipedia.org/wiki/Set_(mathematics)

About this result

Ontological neighbourhood

Mathematics

Mathematical logic: Set theory

Related courses (32)

MATH-410: Riemann surfaces

COM-406: Foundations of Data Science

PHYS-467: Machine learning for physicists

Related lectures (32)

Mapping Functions and Surjections

Explores mapping functions, surjections, injective and surjective functions, and bijective functions.

MATH-106(f) Analyse II Summer 2024 Thursday session week 12

Composition of Applications in Mathematics

Explores the composition of applications in mathematics and the importance of understanding their properties.

Related publications (32)

Self-supervised pre-training enables marine debris detection across sensors

Plastic pollution is a global crisis that needs to be addressed at scale. Current efforts have focused on annotating Sentinel-2 acquisitions for floating debris, and machine learning models are trained to automatically detect such marine debris. However, Sentinel-2 has a revisit time of approximately one week, and images are acquired at 10m resolution, leading to insufficient spatial and temporal resolution for collection efforts. To pave the way for daily monitoring of marine pollution at high spatial and

Elsevier BV2026

Automated Renal Tubule and Cyst Quantification (ARTCyQ)

ARTCyQ is a modular image analysis framework designed for the automated detection, segmentation, and classification of renal tubules and cysts in kidney sections using immunofluorescence imaging. The ARTCyQ pipeline combines:

Deep learning-based segmentation (Omnipose) for tubule detection
Pixel classification (QuPath) for cyst detection, and nuclear and cytoplasmic segmentation
Custom Groovy scripts for automated processing

This repository contains:

A general guideline (1-ARTCyQ_guidelin

Zenodo2026

Adaptive Robust Markov Decision Process for Wide-Area Surveillance with Collaborative Combat Aircraft

Collaborative Combat Aircraft (CCAs) are envisioned to enable autonomous Intelligence, Surveillance, and Reconnaissance (ISR) missions in contested environments, where adversaries may act strategically to deceive or evade detection. These missions pose significant challenges due to model uncertainty and the need for safe, real-time decision-making. While reinforcement learning (RL) offers adaptability, it lacks the safety guarantees required for critical operations. Robust Markov Decision Processes (RMDPs)

American Institute of Aeronautics and Astronautics2026

Related people (31)

Martin Vetterli

Martin Vetterli was appointed president of EPFL by the Federal Council following a selection process conducted by the ETH Board, which unanimously nominated him. Professor Vetterli was born on 4 October 1957 in Solothurn and received his elementary and secondary education in Neuchâtel Canton. He earned a Bachelor’s degree in electrical engineering from ETH Zurich (ETHZ) in 1981, a Master’s of Science degree from Stanford University in 1982, and a PhD from EPFL in 1986. Professor Vetterli taught at Columbia University as an assistant and then associate professor. He was subsequently named full professor in the Department of Electrical Engineering and Computer Sciences at the University of California at Berkeley before returning to EPFL as a full professor at the age of 38. He has also taught at ETHZ and Stanford University. Professor Vetterli has earned numerous national and international awards for his research in electrical engineering, computer science and applied mathematics, including the National Latsis Prize in 1996. He is a fellow of both the Association for Computing Machinery and the Institute of Electrical and Electronics Engineers and a member the US National Academy of Engineering. He has published over 170 articles and three reference works. Professor Vetterli’s work on the theory of wavelets, which are used in signal processing, is considered to be of major importance by his peers, and his areas of expertise, including image and video compression and self-organized communication systems, are central to the development of new information technologies. As the founding director of the National Centre of Competence in Research on Mobile Information and Communication Systems, Professor Vetterli is a staunch advocate of transdisciplinary research. Professor Vetterli knows EPFL inside and out. An EPFL graduate himself, he began been teaching at the school in 1995, was vice president for International Affairs and then Institutional Affairs from 2004 to 2011, and served as dean of the School of Computer and Communication Sciences in 2011 and 2012. In addition to his role as president of the National Research Council of the Swiss National Science Foundation, a position he held from 2013 to 2016, he heads the EPFL’s Audiovisual Communications Laboratory (LCAV) since 1995. Professor Vetterli has supported more than 60 students in Switzerland and the United States in their doctoral work and makes a point of following their highly successful careers, whether it is in the academic or business world. He is the author of some 50 patents, some of which were the basis for start-ups coming out of his lab, such as Dartfish and Illusonic, while others were sold (e.g. Qualcomm) as successful examples of technology transfer. He actively encourages young researchers to market the results of their work.

Florian Karl Richter

Curriculum vitae You can download my CV here ResearchResearch Fields Ergodic Theory, Combinatorics, Number TheoryList of Publications Open-access preprints and e-prints of all my publications are freely available on arXiv ↗ .An inverse theorem for sumsets of sets of positive density in the integersEthan Ackelsberg, Florian K. Richter submitted, 117 pageslink ↗Weighted averages of arithmetic functions and applications to equidistributionVitaly Bergelson, Michael Reilly, Florian K. Richter submitted, 29 page

Samy Bengio

Expertise Large Language ModelsReasoningStatistical machine learningSpeech processingPerson authenticationMultimodal processingSequence processingArtificial neural networks, support vector machinesHidden markov models, gaussian mixture modelsetc Expertise Large Language ModelsReasoningStatistical machine learningSpeech processingPerson authenticationMultimodal processingSequence processingArtificial neural networks, support vector machinesHidden markov models, gaussian mixture modelsetc Mission Samy Bengio

Sabine Süsstrunk

Prof. Dr. Sabine Süsstrunk leads the Image and Visual Representation Lab in the School of Computer and Communication Sciences (IC) at EPFL since 1999. From 2015-2020, she was also the first Director of the Digital Humanities Institute (DHI), College of Humanities (CdH). Her main research areas are in computational photography, computational imaging, color image processing and computer vision, machine learning, and computational image quality and aesthetics. Sabine has authored and co-authored over 200 publications, of which 7 have received best paper/demo awards, and holds over 10 patents. Sabine served as chair and/or committee member in many international conferences on image processing, computer vision, and image systems engineering. She is President of the Swiss Science Council SSC, Founding Member and Member of the Board (President 2014-2018) of the EPFL-WISH (Women in Science and Humanities) Foundation, Member of the Board of the SRG SSR (Swiss Radio and Television Corporation), and Member of the Board of Largo Films. She received the IS&T/SPIE 2013 Electronic Imaging Scientist of the Year Award for her contributions to color imaging, computational photography, and image quality, and the 2018 IS&T Raymond C. Bowman and the 2020 EPFL AGEPoly IC Polysphere Awards for excellence in teaching. Sabine is a Fellow of IEEE and IS&T.

Simon Bossoney

Teaching & PhD Courses Algèbre linéaire et géométrie (pour MAN) PREPA-032(a) We present the fundamental notions of linear algebra in the geometric context where they emerged. We will therefore emphasize the dimensions 2 and 3, and insist on the visualisation of classical objects and operations in the plane and in the 3D space. Analysis I (for CMS) PREPA-003(a) This first semester class is an introduction to the basis of differential calculus. Analysis II (for CMS)

Yiming Li

Sébastien Marcel

Expertise image processing, pattern recognition, machine learning, biometrics Current work Face recognitionSpeaker recognitionMobile biometrics ( http://www.mobioproject.org )Spoofing ( http://www.tabularasa-euproject.org )Biometrics evaluation and testing ( http://www.beat-eu.org ) Sébastien Marcel is the head of the Biometrics group and senior research scientist at the Idiap Research Institute (CH), where he leads a research team and conducts research on biometrics and more particularly face recognition,

Christophe Salzmann

Expertise Real-time Internet ServicesReal-time controlQuality of Service and Bandwidth adaptationLabVIEW Education PhD | Remote experimentation, Network 2005 – 2005 EPFL Directed by Denis Gillet MS. Degree | Computer Science 1999 – 1999 University of Florida Directed by Prof. H. Latchman Teaching & PhD Past EPFL PhD Students as codirector Damien Perritaz, Evgeny Bogdanov, Wissam Halimi Courses Control systems + TP ME

Aude Billard

Expertise RoboticsMachine LearningArtificial IntelligenceHuman-Robot InteractionMechatronicsDynamical Systems ControlComputational Neuroscience Aude Billard is full professor and head of the LASA laboratory at the School of Engineering at the Swiss Institute of Technology Lausanne (EPFL), and the Director of the Swiss Innovation Booster on Robotics. Dr. Billard acts as the President of the IEEE Robotics and Automation Society (RAS), after serving in several roles in the administrative and executive committe

François Fleuret

Related units (32)

L'IDIAP Laboratory

Transportation and Mobility Laboratory

Polymers Laboratory

Related concepts (32)

Set theory

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole. The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory.

Empty set

In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set. Any set other than the empty set is called non-empty. In some textbooks and popularizations, the empty set is referred to as the "null set".

Ordered pair

In mathematics, an ordered pair (a, b) is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (a, b) is different from the ordered pair (b, a) unless a = b. (In contrast, the unordered pair {a, b} equals the unordered pair {b, a}.) Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional vectors.

Related MOOCs (8)

Launching New Ventures

Sharpen your “entrepreneurial mindset”. Launching New Ventures, a course about Entrepreneurship & Strategy for Technology-Driven Startups, is designed for professionals, entrepreneurs, and students who want to learn about how to successfully launch a new business. Through short, animated lectures, inspiring interviews with successful entrepreneurs from around the world, quizzes and suggested readings, students will gain a broad overview of entrepreneurship strategy with insights they can apply to their ven

New Space Economy

The New Space sector is a fast-growing market, boosting the historical institutional space exploration economy. The roughly 366 billion USD global space industry in 2019 is estimated to surge to over $1 trillion by 2040. The current growth in the development of space infrastructures and production of data from space is opening a full range of new applications for new customers. New business models for space encompass launchers, satellite manufacturers, in-orbit servicing, telecommunication, cybersecurity, e

Intro to Traffic Flow Modeling and Intelligent Transport Systems

Travelers in large cities experience significant congestion during their everyday trips. Expanding roads and infrastructure is not a long-lasting remedy to urban congestion. This course will focus on understanding traffic congestion and will explore ways to improve mobility through advanced traffic management schemes. As we all experience, congestion is a highly complex process without a single explanation. An interesting question is: “Can we describe these complex interactions with simple and elegant ways?

Related exercises (13)

Excitation of a vertical harmonic oscillator

A mass m is suspended on a spring of natural length l₀ and of elastic constant k. The spring is attached to a piston. The vertical motion of the piston is described by the equation s(t)u2004=u2004s₀cosu2006(ωt)xa0.

Give explicit expressions for the external forces and determine the equation of motion.
Show that the equation of motion can be recast as, $\ddot{x}\left(t\right) + c_1\,x\left(t\right) = c_1\,s\left(t\right)\ ,$ where x(t)u2004=u2004z(t)u2005+u2005c₂ with z(t) the vertica

The crow and the fox

A few days after the events narrated in the famous fable[1], the crow and the fox meet again. Having learned from its mistakes, the crow is determined not to be fooled again by the fox’s flattery. The fox, however, is set on getting a second piece of cheese. This time , it chooses a more direct method and throws a stone at the crow to scare it. It works beyond expectations: as soon as the stone is thrown, the crow drops the cheese. The crow is perched on a tree of height H; the fox is a distance L from the

Spherical shell

Calculate, in any point of space, the electrostatic potential

V(\bf{r})

and the field

\bf{E}(\bf{r})

created by a spherical shell uniformly charged with an internal radius a, external b, and of total charge u2005+u2005Q, centered at the origin O of the cartesian axes. Graphically represent the functions

V(\bf{r})

and

E_r(\bf{r})

(the radial component of

\bf{E}(\bf{r})

) Replace now the charged shell by an ideal conductor of similar shape. The conductor is neutral and isolated. A charge u2005+u2005

Related notebooks (23)

Density of States in Magnetic Fields Using Landau Levels

Start by setting some parameters including an (artificial) analytical expression of the Fermi surface along kz direction"

"### Define some expressions according to Landau quantization, such as the energy spacing and projection area in k-space"

Set up a histogram to count interceptions between the Fermi surface and "Landau tubes", compute density of states"

Wavepacket construction and propagation in 1D tight-binding model

"# Wavepacket construction and propagation along the 1D chain \n", "\n", "## Introduction\n", "In this notebook, we demonstrate the wavepacket construction and propagation, which is at the heart of the semiclassical approximation. For simplicty, we will consider the familiar 1D chain described using the tight-binding Hamiltonian. We will have to take a sufficiently large number of sites

N

and impose periodic boundary conditions. For simplicity, in the code below we assume lattice constant $a =

Estimation of Electron Density and Mean Free Time for Transparent Conductors

Chapter 1: Transparent Conductors[1]¶ Instruction: The purpose of this notebook is to try and find values of electron density and mean free time for a material suitable to be used as a transparent conductors. The final estimate should be the order of the two parameters to be used[2]¶ In [1]: import numpy as np from bokeh.io import push_notebook, output_notebook, show from bokeh.layouts import row, column from bokeh.plotting import figure output_notebook() from ipywidgets import interact from collect

Modeling Epidemics with Signal Processing Techniques

Epidemics and Signal Processing[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

In [1]: import matplotlib.pyplot as plt import numpy as np import scipy.signal as sp import seaborn as sns sns.set() In [2]: plt.rcParams["figure.figsize"] = (14,4) The basic reproduction number[4]¶ The propagation mechanism of an epidemic such as the one caused by SARS-CoV-2 virus can be roughly modeled, at least in its inital phase, as a process in which each infected individ

Band Structure Investigation of Monolayer and Bilayer Graphene Using the Tight Binding Model

In [1]: import matplotlib.pyplot as plt import numpy as np from mpl_toolkits import mplot3d import itertools %matplotlib widget Band Structure of Mono and Bilayer Graphene[1]¶ If you have questions about this notebook, e-mail Oleg Malanyuk (oleg.malanyuk@epfl.ch) The purpose of this notebook is to investigate the band model of Mono and Bilayer graphene modeled with tight binding model. Tight binding model for monolayer graphene[2]¶ Monolayer graphene is a sheet of carbons aranged in a he