concepts in set theory

The aim of this entry is to present a list of the key objects and concepts used in set theoryMathworldPlanetmath. Each entry in the list links (or will link in the future) to the corresponding PlanetMath entry where the object is presented in greater detail. For convenience, this list also presents the encouraged notation to use (at PlanetMath) for these objects.

  • set

  • set axioms

  • , the empty setMathworldPlanetmath (also {} or ),

  • {x}, singleton,

  • {a1,a2,a3,} (list form), the set with elements a1,a2,a3,,

  • xA, x is an element of the set A,

  • Ax, A is a set containing x,

  • xA, x is not an element of the set A,

  • AB, union of sets A and B,

  • iIAi, union of a family of sets Ai indexed by elements of I,

  • AB, disjoint unionMathworldPlanetmath (or AB),

  • AB, intersection of sets A and B,

  • iIAi, intersectionDlmfMathworldPlanetmath of a family of sets Ai, indexed by elements in I,

  • AB, set differenceMathworldPlanetmath. An alternative notationDlmfDlmfDlmfDlmfDlmf for this is A-B, which should be avoided since in the context of vector spaces, A-B is used for the set of all elements of the form a-b (see Minkowski sum (,

  • A/, set of equivalence classesMathworldPlanetmathPlanetmath in A determined by an equivalence relation in A,

  • [a], equivalence class in A/ generated by aA,

  • A, set complement of A (where the ambient set containing A is understood from context),

  • AB, symmetric set difference of A and B,

  • A×B, Cartesian product of A and B,

  • iIAi, Cartesian product of the sets Ai (sometimes also ×AiiI),

  • idX, identity mapping XX,

  • 𝒫(A), power setMathworldPlanetmath of A (also 2A),

  • BA or BA, the set of functionsMathworldPlanetmath from A to B (rare outside of logic and set theory),

  • f:AB,  f is a function having domain A and codomain B,

  • card(A), cardinality of A (also A or |A|, which can be confused with the absolute valueMathworldPlanetmathPlanetmathPlanetmath),

  • A=B, A and B are equal (generally as sets; occasionally this notation is used to mean “A is canonically isomorphic to B”),

  • AB, A is a subset of B (or AB, especially in set theory and logic),

  • AB, A is a proper subsetMathworldPlanetmathPlanetmath (that is, AB but AB; occasionally authors will use AB to mean “proper subset”, conflicting with the above),

  • AB, A is a supersetMathworldPlanetmath of B (with the same caveats as the previous entries).

  • discrete topology

Set builder notation

{xA such that 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛}.

When A is obvious it may be omitted. Other symbols are also sometimes used in place of the words “such that”, for example



{xA s.t. 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛}.

The reader should take care that if the objects under discussion are not just sets (say, groups or schemes) the operationsMathworldPlanetmath may not be simple set operations, but rather their analogue in the relevant category. For example, the productPlanetmathPlanetmath of two groups is usually assigned a group law of a particular form, while the product of two schemes has “extra” points beyond those obtained from the Cartesian product of the schemes. Such conventions will normally be defined along with the category itself, although occasionally they will be an example of a general notion defined the same way in all categories (such as the categorical direct product).

Title concepts in set theory
Canonical name ConceptsInSetTheory
Date of creation 2013-03-22 14:13:14
Last modified on 2013-03-22 14:13:14
Owner matte (1858)
Last modified by matte (1858)
Numerical id 54
Author matte (1858)
Entry type Topic
Classification msc 03E99
Related topic Set
Related topic PolynomialFunction