concepts in set theory
The aim of this entry is to present a list of the key objects and concepts used in set theory^{}. 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.

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set

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set axioms
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$\mathrm{\varnothing}$, the empty set^{} (also $\{\}$ or $\mathrm{\varnothing}$),

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$\{x\}$, singleton,

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$\{{a}_{1},{a}_{2},{a}_{3},\mathrm{\dots}\}$ (list form), the set with elements ${a}_{1},{a}_{2},{a}_{3},\mathrm{\dots}$,

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$x\in A$, $x$ is an element of the set $A$,

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$A\ni x$, $A$ is a set containing $x$,

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$x\notin A$, $x$ is not an element of the set $A$,

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$A\cup B$, union of sets $A$ and $B$,

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${\bigcup}_{i\in I}{A}_{i}$, union of a family of sets ${A}_{i}$ indexed by elements of $I$,

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$A\coprod B$, disjoint union^{} (or $A\stackrel{\cdot}{\cup}B$),

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$A\cap B$, intersection of sets $A$ and $B$,

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${\bigcap}_{i\in I}{A}_{i}$, intersection^{} of a family of sets ${A}_{i}$, indexed by elements in $I$,

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$A\setminus B$, set difference^{}. An alternative notation^{} for this is $AB$, which should be avoided since in the context of vector spaces, $AB$ is used for the set of all elements of the form $ab$ (see Minkowski sum (http://planetmath.org/MinkowskiSum2)),

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$A{/}_{\sim}$, set of equivalence classes^{} in $A$ determined by an equivalence relation $\sim $ in $A$,

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$[a]$, equivalence class in $A{/}_{\sim}$ generated by $a\in A$,

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${A}^{\mathrm{\complement}}$, set complement of $A$ (where the ambient set containing $A$ is understood from context),

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$A\mathrm{\u25b3}B$, symmetric set difference of $A$ and $B$,

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$A\times B$, Cartesian product of $A$ and $B$,

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${\prod}_{i\in I}{A}_{i}$, Cartesian product of the sets ${A}_{i}$ (sometimes also $\times $${}_{i\in I}A_{i}$),

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${\mathrm{id}}_{X}$, identity mapping $X\to X$,

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$\mathcal{P}(A)$, power set^{} of $A$ (also ${2}^{A}$),

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${}^{A}B$ or ${B}^{A}$, the set of functions^{} from $A$ to $B$ (rare outside of logic and set theory),

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$f:A\to B$, $f$ is a function having domain $A$ and codomain $B$,

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$\mathrm{card}(A)$, cardinality of $A$ (also $\mathrm{\u266f}A$ or $\leftA\right$, which can be confused with the absolute value^{}),

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

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$A\subseteq B$, $A$ is a subset of $B$ (or $A\subset B$, especially in set theory and logic),

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$A\u228aB$, $A$ is a proper subset^{} (that is, $A\subseteq B$ but $A\ne B$; occasionally authors will use $A\subset B$ to mean “proper subset”, conflicting with the above),

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$A\supseteq B$, $A$ is a superset^{} of $B$ (with the same caveats as the previous entries).

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discrete topology
Set builder notation
$$\left\{x\in A\text{such that}\text{\mathit{c}\mathit{o}\mathit{n}\mathit{d}\mathit{i}\mathit{t}\mathit{i}\mathit{o}\mathit{n}}\right\}.$$ 
When $A$ is obvious it may be omitted. Other symbols are also sometimes used in place of the words “such that”, for example
$$\{x\in A\mid \text{\mathit{c}\mathit{o}\mathit{n}\mathit{d}\mathit{i}\mathit{t}\mathit{i}\mathit{o}\mathit{n}}\},$$ 
$$\{x\in A:\text{\mathit{c}\mathit{o}\mathit{n}\mathit{d}\mathit{i}\mathit{t}\mathit{i}\mathit{o}\mathit{n}}\},$$ 
$$\{x\in A\mathrm{\vdots}condition\}$$ 
or
$$\left\{x\in A\text{s.t.}\text{\mathit{c}\mathit{o}\mathit{n}\mathit{d}\mathit{i}\mathit{t}\mathit{i}\mathit{o}\mathit{n}}\right\}.$$ 
The reader should take care that if the objects under discussion are not just sets (say, groups or schemes) the operations^{} may not be simple set operations, but rather their analogue in the relevant category. For example, the product^{} 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  20130322 14:13:14 
Last modified on  20130322 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 