Symbols:Set Theory
Symbols used in Set Theory
Set Delimiters
- $\set {x, y, z}$
Denotes that the objects $x, y, z$ are the elements of a set.
The $\LaTeX$ code for \(\set {x, y, z}\) is \set {x, y, z}
.
Is an Element Of
- $\in$
Let $S$ be a set.
An element of $S$ is a member of $S$.
The $\LaTeX$ code for \(x \in S\) is x \in S
.
Contains as an Element
- $\ni$
$S \ni x$ means that $x$ is an element of the set $S$.
The $\LaTeX$ code for \(S \ni x\) is S \ni x
.
Subset
- $\subseteq$
$S \subseteq T$ means $S$ is a subset of $T$.
In other words, every element of $S$ is also an element of $T$.
Note that this symbol allows the possibility that $S = T$.
The $\LaTeX$ code for \(\subseteq\) is \subseteq
.
Proper Subset
- $\subsetneq$ or $\subsetneqq$
$S \subsetneq T$ and $S \subsetneqq T$ both mean:
- $S$ is a proper subset of $T$
In other words, $S \subseteq T$ and $S \ne T$.
The $\LaTeX$ code for \(\subsetneq\) is \subsetneq
.
The $\LaTeX$ code for \(\subsetneqq\) is \subsetneqq
.
Superset
- $\supseteq$
$S \supseteq T$ means $S$ is a superset of $T$, or equivalently, $T$ is a subset of $S$.
Thus every element of $T$ is also an element of $S$.
Note that this symbol allows the possibility that $S = T$.
The $\LaTeX$ code for \(\supseteq\) is \supseteq
.
Proper Superset
- $\supsetneq$ or $\supsetneqq$
$S \supsetneq T$ and $S \subsetneqq T$ both mean:
- $S$ is a proper superset of $T$
In other words, $S \supseteq T$ and $S \ne T$.
The $\LaTeX$ code for \(\supsetneq\) is \supsetneq
.
The $\LaTeX$ code for \(\supsetneqq\) is \supsetneqq
.
Set Intersection
- $\cap$
$S \cap T$ denotes the intersection of $S$ and $T$.
That is, $S \cap T$ is defined to be the set containing all the elements that are in both the sets $S$ and $T$:
- $S \cap T := \set {x: x \in S \land x \in T}$
The $\LaTeX$ code for \(\cap\) is \cap
.
Set Union
- $\cup$
Let $S$ and $T$ be sets.
The (set) union of $S$ and $T$ is the set $S \cup T$, which consists of all the elements which are contained in either (or both) of $S$ and $T$:
- $x \in S \cup T \iff x \in S \lor x \in T$
The $\LaTeX$ code for \(\cup\) is \cup
.
Empty Set
- $\O$
The empty set:
- $\O = \set {}$
The $\LaTeX$ code for \(\O\) is \O
or \varnothing
or \empty
.
Deprecated
An alternative but less attractive symbol for the empty set is $\emptyset$.
This symbol is deprecated on $\mathsf{Pr} \infty \mathsf{fWiki}$.
The $\LaTeX$ code for \(\emptyset\) is \emptyset
.
Set Difference
- $\setminus$
The difference between two sets $S$ and $T$ is denoted $S \setminus T$ and consists of all the elements of $S$ which are not elements of $T$.
- $S \setminus T := \set {x \in S: x \notin T}$
The $\LaTeX$ code for \(\setminus\) is \setminus
.
Symmetric Difference
- $\symdif$
The symmetric difference between two sets $S$ and $T$ is denoted $S \symdif T$ and consists of all the elements in either set, but not in both.
- $S \symdif T := \paren {S \setminus T} \cup \paren {T \setminus S}$
The $\LaTeX$ code for \(\symdif\) is \symdif
.
Set Complement
Symbols:Set Theory/Set Complement
Cartesian Product
- $\times$
$S \times T$ denotes the Cartesian product of $S$ and $T$.
The $\LaTeX$ code for \(S \times T\) is S \times T
.
Order Sum
- $\oplus$
$S_1 \oplus S_2$ denotes the order sum of two ordered sets $S_1$ and $S_2$.
The $\LaTeX$ code for \(\oplus\) is \oplus
.
Cardinality
- $\card S$
The cardinality of the set $S$.
For finite sets, this means the number of elements in $S$.
The $\LaTeX$ code for \(\card {S}\) is \card {S}
.
Set Equivalence
- $S \sim T$
Let $S$ and $T$ be sets.
Then $S$ and $T$ are equivalent if and only if:
That is, if and only if they have the same cardinality.
This can be written $S \sim T$.
If $S$ and $T$ are not equivalent we write $S \nsim T$.
The $\LaTeX$ code for \(S \sim T\) is S \sim T
.
The $\LaTeX$ code for \(S \nsim T\) is S \nsim T
.
Mapping
A mapping $f \subset A \times B$ can be written:
- $f: A \to B$
or:
- $A \stackrel {f} {\longrightarrow} B$
If $a \in A$ and $b \in B$ such that $\map f a = b$ then we can write:
- $f: a \mapsto b$
If $f$ is an injection this can be written:
- $f: A \rightarrowtail B$ or $f: A \hookrightarrow B$
Similarly a surjection can be written:
- $f: A \twoheadrightarrow B$
Notations for bijection include:
- $f: A \leftrightarrow B$ or $f: A \stackrel {\sim} {\longrightarrow} B$
The $\LaTeX$ code for these symbols are as follows:
- The $\LaTeX$ code for \(f: A \to B\) is
f: A \to B
.
- The $\LaTeX$ code for \(A \stackrel {f} {\longrightarrow} B\) is
A \stackrel {f} {\longrightarrow} B
.
- The $\LaTeX$ code for \(f: a \mapsto b\) is
f: a \mapsto b
.
- The $\LaTeX$ code for \(f: A \rightarrowtail B\) is
f: A \rightarrowtail B
.
- The $\LaTeX$ code for \(f: A \hookrightarrow B\) is
f: A \hookrightarrow B
.
- The $\LaTeX$ code for \(f: A \twoheadrightarrow B\) is
f: A \twoheadrightarrow B
.
- The $\LaTeX$ code for \(f: A \leftrightarrow B\) is
f: A \leftrightarrow B
.
- The $\LaTeX$ code for \(f: A \stackrel {\sim} {\longrightarrow} B\) is
f: A \stackrel {\sim} {\longrightarrow} B
.
Composition of Mappings
- $\circ$
The composite mapping $f_2 \circ f_1$ is defined as:
- $\forall x \in S_1: \map {\paren {f_2 \circ f_1} } x := \map {f_2} {\map {f_1} x}$
The $\LaTeX$ code for \(f \circ g\) is f \circ g
.
Inverse Mapping
- $f^{-1}$
Let $f: S \to T$ be a mapping.
Let $f^{-1} \subseteq T \times S$ be the inverse of $f$:
- $f^{-1} := \set {\tuple {t, s}: \map f s = t}$
Let $f^{-1}$ itself be a mapping:
- $\forall y \in T: \tuple {y, x_1} \in f^{-1} \land \tuple {y, x_2} \in f^{-1} \implies x_1 = x_2$
and
- $\forall y \in T: \exists x \in S: \tuple {y, x} \in f$
Then $f^{-1}$ is called the inverse mapping of $f$.
The $\LaTeX$ code for \(f^{-1}\) is f^{-1}
.
Universal Quantifier
- $\forall$
- $\forall x \in S: \map P x$ means that the propositional function $\map P x$ is true for every $x$ in the set $S$.
- $\forall x: \map P x$ means that the propositional function $\map P x$ is true for every $x$ in the universal set.
The $\LaTeX$ code for \(\forall\) is \forall
.
Existential Quantifier
- $\exists$
- $\exists x \in S: \map P x$ means that there exists at least one $x$ in the set $S$ for which the propositional function $\map P x$ is true.
- $\exists x: \map P x$ means that there exists at least one $x$ in the universal set for which the propositional function $\map P x$ is true.
The $\LaTeX$ code for \(\exists\) is \exists
.
Negation
- $\not \in, \not \exists, \not \subseteq, \not \subset, \not \supseteq, \not \supset$
The above symbols all mean the opposite of the non struck through version of the symbol.
For example, $x \not \in S$ means that $x$ is not an element of $S$.
The slash through a symbol ($/$) can be used to reverse the meaning of essentially any mathematical symbol (especially relations), although it is used most frequently with those listed above.
Note that $\not \subsetneq$ and $\not \supsetneq$ can be confusing due to the strike through of the symbol as a whole and the strike through of the equivalence bar on the bottom, and hence should likely be avoided.
The $\LaTeX$ code for negation is \not
followed by the code for whatever symbol you want to negate.
For example, \not \in
will render $\not\in$.
Variants
Set Difference
- $-$
A variant notation for the difference between two sets $S$ and $T$ is $S - T$.
The $\LaTeX$ code for \(S - T\) is S - T
.
Symmetric Difference
There is no standard symbol for symmetric difference. The one used here, and in general on $\mathsf{Pr} \infty \mathsf{fWiki}$:
- $S \symdif T$
is the one used in 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics.
The following are often found for $S \symdif T$:
- $S * T$
- $S \oplus T$
- $S + T$
- $S \mathop \triangle T$
According to 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics: symmetric difference:
- $S \mathop \Theta T$
- $S \mathop \triangledown T$
are also variants for denoting this concept.
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.): symmetric difference recognizes a further variant:
- $S \mathop \nabla T$
Deprecated Symbols
This page contains symbols which may or may not be in current use, but are either non-standard in mathematics or have been superseded by their more modern variants.
However, on $\mathsf{Pr} \infty \mathsf{fWiki}$ the intention is to present a consistent style, and so these symbols are to be considered deprecated.
Subset
$\subset$ is sometimes used to mean:
- $S$ is a subset of $T$
in the sense that $S$ is permitted to equal $T$.
That is, for which we have specified as $S \subseteq T$.
Although many sources use this interpretation, it is emphatically not recommended, as it can be the cause of considerable confusion.
Superset
$\supset$ is sometimes used to mean:
- $S$ is a superset of $T$
in the sense that $S$ is permitted to equal $T$.
That is, for which we have specified as $S \supseteq T$.
Although many sources use this interpretation, it is emphatically not recommended, as it can be the cause of considerable confusion.