41 articles
Barber paradox
The barber paradox is a logic puzzle asking whether a barber who shaves all and only men who do not shave themselves shaves himself, since no such barber can exist.
Binary relation
A binary relation, also known as a dyadic or two-place relation, is a set of ordered pairs recording which elements of one set relate to which elements of another.
Cardinal number
A cardinal number is a number that measures the size of a set by counting its elements; Georg Cantor developed the concept in the nineteenth century.
Cardinality
Cardinality is the size of a set, written |A|, defined through one-to-one correspondence. Georg Cantor showed in 1873 that infinite sets come in different sizes.
Cartesian product
The Cartesian product of two sets A and B, written A × B, is the set of all ordered pairs (a, b) with a in A and b in B.
Complement (set theory)
In set theory, the complement of a set is the elements, drawn from a fixed universe, that do not belong to the set, such as A \ B.
Countable set
A countable set is a set whose elements can be matched one-to-one with some or all of the natural numbers, a concept introduced by Georg Cantor.
Dichotomy
A dichotomy is a division of a whole into exactly two mutually exclusive, jointly exhaustive parts. In logic it divides a genus into two species by a single attribute.
Disjoint sets
Disjoint sets, also called mutually exclusive sets, are sets that share no elements, so their intersection is empty; the concept underlies partitions and counting rules.
Domain of a function
The domain of a function, also called the function domain, is the set of inputs for which it is defined, the X in f: X → Y.
Element of a set
An element of a set, also called a member, is an object that belongs to the set, written a ∈ A; membership is the basic relation of set theory.
Empty set
The empty set, or empty subset, is the unique set containing no elements, denoted ∅; it has cardinality zero and is a subset of every set.
Equivalence class
An equivalence class is the subset collecting all elements related to a given element by an equivalence relation; the classes form a partition of the set.
Euler diagram
An Euler diagram represents sets as labeled closed curves whose spatial arrangement shows containment, intersection, and exclusion. Euler introduced it in 1761, and unlike Venn diagrams it draws only relevant intersections.
Finite set
A finite set is a set whose elements can be counted with a natural number n; the number n is the set's cardinality, and any set that is not finite is infinite.
Georg Cantor
Georg Cantor, in full Georg Ferdinand Ludwig Philipp Cantor (1845–1918), was a German mathematician at the University of Halle who created set theory as a mathematical discipline.
Hilbert's paradox of the Grand Hotel
Hilbert's paradox of the Grand Hotel, also known as Hilbert's Hotel, is a thought experiment in which a fully occupied infinite hotel still makes room for new guests.
Image (mathematics)
In mathematics, the image of a function is the set of all outputs it produces, and the image of a set under a function collects those outputs.
Indicator function
The indicator function, also called the characteristic function of a set, maps elements of a set to 1 if they belong to a subset and 0 otherwise.
Intersection (set theory)
In set theory, the intersection of sets A and B, written A ∩ B, is the set of elements they share, the set-level counterpart of logical conjunction.
Inverse function
An inverse function is a function that reverses another function, sending each output back to its input. It exists only when the function is bijective.
Multiset
A multiset, also called a bag, is a mathematical collection in which elements may appear more than once, each carrying a multiplicity, unlike an ordinary set where repeats collapse.
Ordered pair
An ordered pair (a, b) is a mathematical object with a first and second coordinate, so (a, b) equals (c, d) only when a = c and b = d.
Power set
The power set of a set A, written P(A) or 2^A, is the set of all subsets of A, always including the empty set and A itself.
Pullback (category theory)
A pullback, also called the fibered product, is the limit in category theory of two morphisms with a common codomain; in sets it is the set of pairs (a, b) with f(a) = g(b).
Quotient set
A quotient set is the set of all equivalence classes of a set under an equivalence relation, so equivalence becomes equality, as in the integers modulo n.
Reflexive relation
A reflexive relation, also called the reflexive property, is a binary relation on a set in which every element is related to itself, as with the identity relation.
Set (mathematics)
A set is a well-determined collection of distinct objects, called its members or elements, treated as a single mathematical object; set theory is the standard foundation of mathematics.
Set theory
Set theory is the branch of mathematics that studies collections of objects called sets, founded by Georg Cantor in the 1870s and now the standard foundation of modern mathematics.
Set-builder notation
Set-builder notation, also called set comprehension, is the mathematical notation for specifying a set by a property its members satisfy, as in {x ∈ A : P(x)}.