In mathematics, the empty set or void 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 mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their intersection is the point at which they meet. More generally, in set theory, the intersection of sets is defined to be the set of elements which belong to all of them.
Intersections can be thought of either collectively or individually, see Intersection (geometry) for an example of the latter. The definition given above exemplifies the collective view, whereby the intersection operation always results in a well-defined and unique, although possibly empty, set of mathematical objects. In contrast, the individual view focuses on the separate members of this set. Given this view, intersections need not be unique, as shown by the two points of intersection between a circle and a line pictured. Similarly, (individual) intersections need not exist as between two parallel but distinct lines in Euclidean geometry.
In mathematics, a set is a collection of different things; the things are elements or members of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, or other sets. A set may be finite or infinite. There is a unique set with no elements, called the empty set; a set with a single element is a singleton.
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.
In mathematical analysis, a domain or region is a non-empty, connected, and open set in a topological space. In particular, it is any non-empty connected open subset of the real coordinate space R or the complex coordinate space C. A connected open subset of coordinate space is frequently used for the domain of a function.
The basic idea of a connected subset of a space dates from the 19th century, but precise definitions vary slightly from generation to generation, author to author, and edition to edition, as concepts developed and terms were translated between German, French, and English works. In English, some authors use the term domain, some use the term region, some use both terms interchangeably, and some define the two terms slightly differently; some avoid ambiguity by sticking with a phrase such as non-empty connected open subset.
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general.
The intuition behind this concept dates back to Ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Constantin Carathéodory, and Maurice Fréchet, among others. According to Thomas W. Hawkins Jr., "It was primarily through the theory of multiple integrals and, in particular the work of Camille Jordan that the importance of the notion of measurability was first recognized."
The concept of "The Void" in philosophy encompasses the ideas of nothingness and emptiness, a notion that has been interpreted and debated across various schools of metaphysics. In ancient Greek philosophy, the Void was discussed by thinkers like Democritus, who saw it as a necessary space for atoms to move, thereby enabling the existence of matter. Contrasting this, Aristotle famously denied the existence of a true Void, arguing that nature inherently avoids a vacuum.
In Eastern philosophical traditions, the Void takes on significant spiritual and metaphysical meanings. In Buddhism, Śūnyatā refers to the emptiness inherent in all things, a fundamental concept in understanding the nature of reality. In Taoism, the Void is represented by Wuji, the undifferentiated state from which all existence emerges, embodying both the potential for creation and the absence of form.
In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset.
Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is sometimes called a setoid, typically in type theory and proof theory.
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that distinguish topological spaces.