Partially ordered setMathematical set with an orderingY indicates that the column's property is always true for the row's term (at the very left), while ✗ indicates that the property is not guaranteedin general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric,is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.
Order topologyIn mathematics, an order topology is a specific topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets.
Converse relationIn mathematics, the converse of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent of'.Well-orderY indicates that the column's property is always true for the row's term (at the very left), while ✗ indicates that the property is not guaranteedin general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric,is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.
SubsetIn mathematics, a setA is a subset of a set Bif and only if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment). A is a subset of B may also be expressed as B includes (or contains) A or A is included (or contained) in B. A k-subset is a subset with k elements.
Ordinal numberGeneralization of "n-th" to infinite casesRepresentation of the ordinal numbers up to ωω{\displaystyle \omega ^{\omega }}. One turn of the spiral corresponds to the mapping f(α)=ω(1+α){\displaystyle f(\alpha )=\omega (1+\alpha )}.Binary relationRelationship between elements of two setsY indicates that the column's property is always true for the row's term (at the very left), while ✗ indicates that the property is not guaranteedin general (it might, or might not, hold). For example, that every equivalence relation is symmetric, but not necessarily antisymmetric,is indicated by Y in the "Symmetric" column and ✗ in the "Antisymmetric" column, respectively.
Set (mathematics)In mathematics, a set is a collection of different things; the things are called elements or members of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, functions, or even other sets. Sets cannot be mathematically defined, since they form a primitive notion. Instead, they are characterized by basic properties (axioms) that sets must have to correspond to intuition about collections.
Upper and lower boundsIn mathematics, particularly in order theory, an upper bound or majorant of a subsetS of some preordered set(K, ≤) is an element of K that is greater than or equal to every element of S. Dually, a lower bound or minorant of S is defined to be an element of K that is less than or equal to every element of S. A set with an upper (respectively, lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound.
Linear extensionIn order theory, a branch of mathematics, a linear extension of a partial order is a total order (or linear order) that is compatible with the partial order. As a classic example, the lexicographic order of totally ordered sets is a linear extension of their product order. DefinitionsLinear extension of a partial orderA partial order is a reflexive, transitive and antisymmetric relation.
MathematicsField of knowledgeIlluminated letter P at the beginning of Adelard of Bath's translation of Euclid's Elements, which starts "Punctum est illud cui pars non est" ('A point is that which has no part'). Geometry is shown personified as a woman, following Martianus Capella's De nuptiis Philologiae et Mercurii.Mathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities.*As an Amazon Associate I earn from qualifying purchases.