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Lecture

Real Numbers: Sets and Operations

Description

This lecture covers the fundamental concepts related to real numbers, including sets, operations, and properties such as supremum and infimum. It explores the definitions and axioms of real numbers, as well as the concepts of supremum and infimum in the context of subsets of real numbers.

Official source

Ontological neighbourhood

Mathematics

Topology: General topology

Mathematical logic: Set theory

Related concepts (32)

Real number

In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives.

Definable real number

Informally, a definable real number is a real number that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of a formal language. For example, the positive square root of 2, , can be defined as the unique positive solution to the equation , and it can be constructed with a compass and straightedge. Different choices of a formal language or its interpretation give rise to different notions of definability.

Completeness of the real numbers

Completeness is a property of the real numbers that, intuitively, implies that there are no "gaps" (in Dedekind's terminology) or "missing points" in the real number line. This contrasts with the rational numbers, whose corresponding number line has a "gap" at each irrational value. In the decimal number system, completeness is equivalent to the statement that any infinite string of decimal digits is actually a decimal representation for some real number.

Projectively extended real line

In real analysis, the projectively extended real line (also called the one-point compactification of the real line), is the extension of the set of the real numbers, , by a point denoted ∞. It is thus the set with the standard arithmetic operations extended where possible, and is sometimes denoted by or The added point is called the point at infinity, because it is considered as a neighbour of both ends of the real line. More precisely, the point at infinity is the limit of every sequence of real numbers whose absolute values are increasing and unbounded.

Meagre set

In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is called nonmeagre, or of the second category. See below for definitions of other related terms. The meagre subsets of a fixed space form a σ-ideal of subsets; that is, any subset of a meagre set is meagre, and the union of countably many meagre sets is meagre.

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Real Numbers: Sets and Operations

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Covers the basics of sets and operations in mathematics, from set properties to advanced operations.

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Covers the concepts of infimum and supremum of subsets in real numbers.

Real Numbers: Sets and Operations

Covers the basics of real numbers and set theory, including subsets, intersections, unions, and set operations.

Exemple: Infimum and Supremum MOOC: Analysis I MOOC: Analysis I (part 3) : Real number sequences I and II

Illustrates the calculation of infimum and supremum for a set of elements.

Related concepts (32)

Real number

Definable real number

Completeness of the real numbers

Projectively extended real line

Meagre set